Ars Inveniendi Analytica (2021), Paper No. 5, 35 pp.

DOI 10.15781/hk9g-zz18

00footnotetext:  Licensed under a Creative Commons Attribution License (CC-
BY).

A Liouville-type theorem for
stable minimal hypersurfaces

Leon Simon
Stanford University

Communicated by Guido De Philippis

Abstract. We prove that if $M$ is a strictly stable complete minimal
hypersurface in $\smash{\varmathbb{R}^{n+1+\ell}}$ which has finite density at
infinity and which lies on one side of a cylinder
$\smash{\varmathbb{C}=\varmathbb{C}_{0}\times\varmathbb{R}^{\ell}}$, where
$\smash{\varmathbb{C}_{0}}$ is a strictly stable area minimizing hypercone in
$\smash{\varmathbb{R}^{n+1}}$ with ${\rm sing\,\,}\varmathbb{C}_{0}=\\{0\\}$,
then $M$ must be cylindrical—i.e. $M=S\times\varmathbb{R}^{\ell}$, where
$S\subset\smash{\varmathbb{R}^{n+1}}$ is a smooth strictly stable minimal
hypersurface in $\,\smash{\varmathbb{R}^{n+1}}$. Applications will be given
in  _[Sim21a], [Sim21b]_.

Keywords. minimal hypersurface, strict stability, minimizing cone

## Introduction

The main theorem here (Theorem 1) establishes that if $M$ is a complete
strictly stable minimal hypersurface in $\varmathbb{R}^{n+1+\ell}$ lying on
one side of a cylindrical hypercone
$\varmathbb{C}=\varmathbb{C}_{0}\times\varmathbb{R}^{\ell}$ with ${\rm
sing\,\,}\varmathbb{C}_{0}=\\{0\\}$ and $\varmathbb{C}_{0}$ strictly stable
and minimizing, then $M$ must itself be cylindrical—i.e. of the form
$S\times\varmathbb{R}^{\ell}$ where $S$ is a smooth complete (not necessarily
connected) strictly stable minimal hypersurface in $\varmathbb{R}^{n+1}$.

This result, or more correctly its Corollary 1 below, is a crucial ingredient
in the author’s recent proof (in [Sim21b]) that, with respect to a suitable
$C^{\infty}$ metric for $\varmathbb{R}^{n+1+\ell}$, $n+\ell\geqslant 8$, there
are examples of strictly stable minimal hypersurfaces which have singular set
of the form $\\{0\\}\times K$, where $K$ is an arbitrary closed subset of
$\varmathbb{R}^{\ell}$.

An outline of the present paper is as follows. After a description of the main
results in §1 and some notational and technical preliminaries in §2 and §3, in
§4 we establish $L^{2}$ growth estimates for solutions of the Jacobi equation
(i.e. the linearization of the minimal surface equation) on $M$. These
estimates are applied to give growth estimates on (i) $(x,y)\cdot\nu$, where
$\nu$ is the unit normal of $M$, (ii)
$\nu_{\\!y}=(e_{n+2}\cdot\nu,\ldots,e_{n+1+\ell}\cdot\nu)$ (i.e. the
components of the unit normal $\nu$ of $M$ in the $y$-coordinate directions),
and (iii) (in §5) $d|M$, where $d(x)={\rm dist\,}((x,y),\varmathbb{C})$ is the
distance to the cylinder $\varmathbb{C}$. In §6 the growth estimates
established in §4 and §5 are combined to show that, if $\nu_{\\!y}$ is not
identically zero,

$R^{\gamma-\alpha}\leqslant\int_{M\cap\\{(x,y):|(x,y)|<R\\}}\nu_{\\!y}^{2}\,d\mu\leqslant
R^{-2+\gamma+\alpha},\,\,\gamma=\ell+2+\beta_{1},\,\,$

for each $\alpha\in(0,1)$ and all sufficiently large $R$ (depending on
$\alpha$ and $M$), where $\mu$ is $(n+\ell)$-dimensional Hausdorff measure,
and $\beta_{1}=2(({\textstyle\frac{n-2}{2}})^{2}+\lambda_{1})^{1/2}$, with
$\lambda_{1}$ the first eigenvalue of the Jacobi operator of the compact
minimal surface $\Sigma=\varmathbb{C}_{0}\cap\varmathbb{S}^{n}$. These
inequalities are clearly impossible for $R>1$, so we finally conclude that
indeed $\nu_{\\!y}$ is identically zero, showing that $M$ is cylindrical as
claimed in the main theorem.

##  1\. Main Results

Let $\varmathbb{C}_{0}$ be a minimal hypercone in $\varmathbb{R}^{n+1}$ with
${\rm
sing\,\,}\varmathbb{C}_{0}=\overline{\varmathbb}{C}_{0}\setminus\varmathbb{C}_{0}=\\{0\\}$
and let $\mathcal{L}_{\varmathbb{C}_{0}}$ be the Jacobi operator (linearized
minimal surface operator) defined by

$\mathcal{L}_{\varmathbb{C}_{0}}u=\Delta_{\varmathbb{C}_{0}}u+|A_{\varmathbb{C}_{0}}|^{2}u,$

where $|A_{\varmathbb{C}_{0}}|^{2}$ is the squared length of the second
fundamental form of $\varmathbb{C}_{0}$. In terms of the Jacobi operator

1.1 $\mathcal{L}_{\Sigma}=\Delta_{\Sigma}u+|A_{\Sigma}|^{2}u$

of the compact submanifold $\Sigma=\varmathbb{C}_{0}\cap\varmathbb{S}^{n}$
(which is minimal in $\varmathbb{S}^{n}$), we have

1.2 $\mathcal{L}_{\varmathbb{C}_{0}}u=r^{1-n}\frac{\partial}{\partial
r}\bigl{(}r^{n-1}\frac{\partial u}{\partial
r}\bigr{)}+r^{-2}\mathcal{L}_{\Sigma},\quad r=|x|.$

If $\lambda_{1}$ is the first eigenvalue of $-\mathcal{L}_{\Sigma}$, and
$\varphi_{1}>0$ is the corresponding eigenfunction (unique up to a constant
factor) we have (possibly complex-valued) solutions
$r^{\gamma_{1}^{\pm}}\varphi_{1}$ of $\mathcal{L}_{\varmathbb{C}_{0}}u=0$,
where $\gamma_{1}^{\pm}$ are the “characteristic exponents”

1.3
$\gamma_{1}^{\pm}=-\frac{n-2}{2}\pm\sqrt{\bigl{(}\frac{n-2}{2}\bigr{)}^{2}+\lambda_{1}}.$

We assume $\varmathbb{C}_{0}$ is strictly stable, i.e. there is $\lambda>0$
with

1.4
$\lambda\int_{\varmathbb{C}_{0}}|x|^{-2}\zeta^{2}(x)\,d\mu(x)\leqslant\int_{\varmathbb{C}_{0}}\bigl{(}|\nabla_{\varmathbb{C}_{0}}\zeta|^{2}-|A_{\varmathbb{C}_{0}}|^{2}\zeta^{2}\bigr{)}\,d\mu\,\,\,\,\forall\zeta\in
C_{c}^{\infty}(\varmathbb{R}^{n+1}),$

where $\mu$ is $n$-dimensional Hausdorff measure; subsequently $\mu$ will
always denote Hausdorff measure of the appropriate dimension.

Using 1.2 and 1.3, 1.4 is readily checked to be equivalent to the condition
that

1.5 $\lambda_{1}>-((n-2)/2)^{2},$

in which case

1.6 $\gamma_{1}^{-}<-\frac{n-2}{2}<\gamma_{1}^{+}<0.$

The main theorem here relates to hypersurfaces
$M\subset\varmathbb{R}^{n+1+\ell}$, where $\ell\geqslant 1$; points in
$\varmathbb{R}^{n+1+\ell}=\varmathbb{R}^{n+1}\times\varmathbb{R}^{\ell}$ will
be denoted

$(x,y)=(x_{1},\ldots,x_{n+1},y_{1},\ldots,y_{\ell}).$

In the main theorem, which we now state and which will be proved in §6, we
assume that (i) $M$ is a smooth complete minimal hypersurface in
$\varmathbb{R}^{n+1+\ell}$ lying on one side of
$\varmathbb{C}=\varmathbb{C}_{0}\times\varmathbb{R}^{\ell}$, i.e.

1.7 $M\subset U_{+}\times\varmathbb{R}^{\ell},$

where $U_{+}$ is one of the two connected components of
$\varmathbb{R}^{n+1+\ell}\setminus\overline{C}_{0}$, that (ii) $M$ is strictly
stable in the sense that (Cf. 1.4) there is $\lambda>0$ with

1.8
$\lambda\int_{M}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant\int_{M}\bigl{(}|\nabla_{M}\zeta|^{2}-|A_{M}|^{2}\zeta^{2}\bigr{)}\,d\mu\,\,\,\,$

for all $\zeta\in C_{c}^{1}(\varmathbb{R}^{n+1+\ell})$, and that (iii) $M$ has
finite density at $\infty$, i.e.

1.9 $\sup_{R>1}R^{-n-\ell}\mu(M\cap B_{R})<\infty,$

where, here and subsequently, $B_{R}$ is the closed ball of radius $R$ and
centre $(0,0)$ in $\varmathbb{R}^{n+1+\ell}$:

$B_{R}=\\{(x,y)\in\varmathbb{R}^{n+1+\ell}:|(x,y)|\leqslant R\\};$

the corresponding open ball, which we shall occasionally use, is denoted

$\breve{B}_{R}=\\{(x,y)\in\varmathbb{R}^{n+1+\ell}:|(x,y)|<R\\}.$

1.10 Theorem (Liouville-type Theorem.) If $M$ is a smooth, complete, embedded
minimal hypersurface (without boundary) in $\varmathbb{R}^{n+1+\ell}$ such
that 1.7, 1.8 and 1.9 hold, then $M$ is cylindrical, i.e.

$M=S\times\varmathbb{R}^{\ell},$

where $S$ is a smooth complete strictly stable minimal hypersurface which is
contained in $U_{+}$.

Remark:  Using the regularity theory of [SS81], the conclusion continues to
hold, with no essential change in the proof, in case $M$ is allowed to have a
singular set of finite $(n+\ell-2)$-dimensional Hausdorff measure. Indeed if
we use the regularity theory of [Wic14] then we need only assume _a priori_
that $M$ has zero $(n+\ell-1)$-dimensional Hausdorff measure.

For applications in [Sim21a], [Sim21b] we now state a corollary of the above
theorem, in which the stability hypothesis on $M$ is dropped, and instead
$\varmathbb{C}_{0}$ is assumed to be both strictly stable and strictly
minimizing, and we impose an _a priori_ smallness assumption on the last
$\ell$ components of the unit normal $\nu_{M}=(\nu_{1},\ldots,\nu_{n+1+\ell})$
of $M$; i.e. a smallness assumption on $(\nu_{n+2},\ldots,\nu_{n+1+\ell})\,$
which we subsequently write as

1.11 $\nu_{y}=(\nu_{y_{1}},\ldots,\nu_{y_{\ell}});\,\,\text{ i.e.\
}\nu_{y_{j}}=e_{n+1+j}\cdot\nu_{M},\,\,\,j=1,\ldots,\ell.$

Notice that such a smallness assumption on $|\nu_{y}|$ amounts to a
restriction on how rapidly $M$ can vary in the $y$-directions.

Recall (see [HS85]) $\varmathbb{C}_{0}$ is said to be strictly minimizing if
there is a constant $c>0$ such that

1.12 $\displaystyle\mu(\varmathbb{C}_{0}\cap B_{1})\leqslant\mu(T\cap
B_{1})-c\rho^{n}\text{ whenever $\rho\in(0,{\textstyle\frac{1}{2}}]$ and $T$
is a smooth }$ $\displaystyle\hskip 36.135pt\text{compact hypersurface-with-
boundary in $B_{1}\setminus B_{\rho}$ with $\partial
T=\varmathbb{C}_{0}\cap\varmathbb{S}^{n}$}.$

1.13 Corollary. Suppose $\varmathbb{C}_{0}$ is strictly stable and strictly
minimizing (as in 1.12) and $\alpha\in(0,1)$. Then there is
$\varepsilon_{0}=\varepsilon_{0}(\varmathbb{C}_{0},\alpha)\in(0,{\textstyle\frac{1}{2}}]$
such that if $M$ is a smooth, complete, embedded minimal hypersurface in
$\varmathbb{R}^{n+1+\ell}$ with

$\left\\{\begin{aligned} &M\subset U_{+}\times\varmathbb{R}^{\ell},\\\
\vskip-2.0pt\cr&{\sup}_{R>1}R^{-n-\ell}\mu(M\cap
B_{R})\leqslant(2-\alpha)\mu(\varmathbb{C}\cap B_{1}),and\\\
\vskip-2.0pt\cr&{\sup}_{M}|\nu_{y}|<\varepsilon_{0},\end{aligned}\right.$

then

$M=\lambda S\times\varmathbb{R}^{\ell}$

for some $\lambda>0$, where $S$ is the minimal hypersurface in $U_{+}$ as in
_[HS85]_ (see 7.1 in §7 below).

Applications of the above results are given in [Sim21a] and [Sim21b]. Although
the assumptions of strict stability in Theorem 1 and $|\nu_{y}|$ small in
Corollary 1 are appropriate for the applications in [Sim21a], [Sim21b], it
would be of interest to know if these restrictions can be significantly
relaxed—for example the question of whether or not mere stability would
suffice in place of the strict stability assumption in Theorem 1.

##  2\. Preliminaries concerning $M$

As in §1, $\varmathbb{C}_{0}$ will be a smooth embedded minimal hypercone in
$\varmathbb{R}^{n+1}$ with ${\rm sing\,\,}\varmathbb{C}_{0}=\\{0\\}$, and we
let $U_{\pm}$ be the two connected components of
$\varmathbb{R}^{n+1}\setminus\overline{\varmathbb{C}}_{0}$.

$M$ will be a smooth embedded minimal hypersurface, usually contained in
$U_{+}\times\varmathbb{R}^{\ell}$, although some results are formulated to
apply locally in a ball, and also independent of the inclusion assumption
$M\subset U_{+}\times\varmathbb{R}^{\ell}$.

For the moment we assume only that $M$ is minimal (i.e. has first variation
zero) in $\breve{B}_{R}$. Thus

2.1 $\int_{M}{\rm
div}_{M}Z(x,y)\,d\mu(x,y)=0,\,\,Z=(Z_{1},\ldots,Z_{n+1+\ell}),\,Z_{j}\in
C^{1}_{c}(\breve{B}_{R}),$

where ${\rm div}_{M}Z$ is the “tangential divergence” of $Z$:

$\displaystyle{\rm
div}_{M}Z_{|(x,y)}={\textstyle\sum}_{j=1}^{n+\ell}\tau_{j}\cdot D_{\tau_{j}}Z$
$\displaystyle={\textstyle\sum}_{k,m=1}^{n+1+\ell}{\textstyle\sum}_{j=1}^{n+\ell}\tau_{j}^{k}\tau_{j}^{m}D_{k}Z_{m}$
$\displaystyle={\textstyle\sum}_{i,j=1}^{n+1+\ell}g^{ij}(x,y)D_{i}Z_{j}(x,y),$

with $\tau_{j}=(\tau_{j}^{1},\ldots,\tau_{j}^{n+1+\ell})$,
$j=1,\ldots,n+\ell$, any locally defined orthonormal basis of $T_{(x,y)}M$,
$(x,y)\in M$ and

2.2
$g^{ij}=\delta_{ij}-\nu_{i}\nu_{j},\quad\nu=(\nu_{1},\ldots,\nu_{n+1+\ell})\text{
a unit normal for }M.$

For $v\in C^{2}(M)$ we let ${\rm graph\,}v$ be the graph of $v$ taken off $M$:

${\rm graph\,}v=\bigl{\\{}(x,y)+v(x,y)\nu_{\\!M}(x,y):(x,y)\in M\bigr{\\}}$

(notice that this may fail to be an embedded hypersurface unless $v$ has small
enough $C^{2}$ norm), and we take

2.3 $\mathcal{M}_{\\!M}(v)=\text{ the mean curvature operator on $M$}.$

Thus $\mathcal{M}_{\\!M}(v)$ is the Euler-Lagrange operator of the area
functional $\mathcal{A}_{\\!M}$ on $M$, defined by

$\mathcal{A}_{\\!M}(v)=\int_{M}J_{M}(V)\,d\mu$

where $V(x,y)=(x,y)+v(x,y)\nu_{M}(x,y)$ is the graph map taking $M$ to ${\rm
graph\,}v$ and $J_{M}(V)$ is the Jacobian of $V$:

$J_{M}(V)=\sqrt{\det\bigl{(}D_{\tau_{i}}V\cdot
D_{\tau_{j}}V\bigr{)}}=\sqrt{\det\bigl{(}\delta_{ij}+v_{i}v_{j}+v^{2}{\textstyle\sum}_{k}h_{ik}h_{jk}+2vh_{ij}\bigr{)}},$

where, for $(x,y)\in M$, $\tau_{1},\ldots,\tau_{n+\ell}$ is an orthonormal
basis for $T_{(x,y)}M$, $h_{ij}$ is the second fundamental form of $M$ with
respect to this basis, and $v_{i}=D_{\tau_{i}}v$. Since
$\sum_{i=1}^{n+\ell}h_{ii}=0$ we then have

2.4
$J_{M}(V)=\sqrt{1+|\nabla_{M}v|^{2}-|A_{M}|^{2}v^{2}+E\bigl{(}v(h_{ij}),v^{2}({\textstyle\sum}_{k}h_{ik}h_{jk}),(v_{i}v_{j})\bigr{)}}$

where $E$ is a polynomial of degree $n+\ell$ in the indicated variables with
constant coefficients depending only on $n,\ell$ and with each non-zero term
having at least degree $3$. So the second variation of $\mathcal{A}_{M}$ is
given by

${\textstyle\frac{d^{2}}{dt^{2}}}\bigl{|}_{t=0}\mathcal{A}(t\zeta)=\int_{M}\bigl{(}|\nabla_{M}\zeta|^{2}-|A_{M}|^{2}\zeta^{2}\bigr{)}\,d\mu=-\int_{M}\zeta\mathcal{L}_{M}\zeta\,d\mu,\quad\zeta\in
C^{1}_{c}(\varmathbb{R}^{n+1+\ell}),$

where $\mathcal{L}_{M}$ is the Jacobi operator on $M$ defined by

2.5 $\mathcal{L}_{M}v=\Delta_{M}v+|A_{M}|^{2}v,$

with $\Delta_{M}f={\rm div}_{M}(\nabla_{M}f)$ the Laplace-Beltrami operator on
$M$ and $|A_{M}|=\bigl{(}\sum h_{ij}^{2}\bigr{)}^{1/2}$ (the length of the
second fundamental form of $M$). Of course, by 2.3 and 2.4, $\mathcal{L}_{M}$
is the linearization of the mean curvature operator at $0$, i.e.

2.6
$\mathcal{L}_{M}v={\textstyle\frac{d}{dt}}\bigl{|}_{t=0}\mathcal{M}_{M}(tv).$

In view of 2.4 and the definition of $\mathcal{L}_{M}$, if
$|A_{M}||u|+|\nabla_{M}u|<1$ on a domain in $M$ then we can write

2.7 $\mathcal{M}(u)=\mathcal{L}_{M}(u)+{\rm div}_{M}E(u)+F(u)$

on that domain, where $|E(u)|\leqslant
C(|A_{M}|^{2}|u|^{2}+|\nabla_{M}u|^{2})$ and $|F(u)|\leqslant
C(|A_{M}|^{3}v^{2}+|A_{M}||\nabla_{M}v|^{2})$, $C=C(n,\ell)$.

We say that $M$ is _strictly stable_ in $\breve{B}_{R}$ if the second
variation of $\mathcal{A}_{\\!M}$ is strictly positive in the sense that

${\inf}_{\zeta\in
C^{1}_{c}(\breve{B}_{R}),\,\int_{M}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)=1}\,\,\,{\textstyle\frac{d^{2}}{dt^{2}}}\bigl{|}_{t=0}\mathcal{A}(t\zeta)>0,$

or, equivalently, that there is $\lambda>0$ such that 1.8 holds.

Observe that if we have such strict stability then by replacing $\zeta$ in 1.8
with $\zeta w$ we obtain

$\displaystyle\smash[b]{\lambda\int_{M}|x|^{-2}w^{2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant\int_{M}\bigl{(}w^{2}|\nabla_{M}\zeta|^{2}+\zeta^{2}|\nabla_{M}w|^{2}}$
$\displaystyle\hskip 180.67499pt+2\zeta
w\nabla_{M}\zeta\cdot\nabla_{M}w-|A_{M}|^{2}\zeta^{2}w^{2}\bigr{)}\,d\mu,$

and $\int_{M}2\zeta w\nabla_{M}\zeta\cdot\nabla_{M}w=\int
w\nabla_{M}\zeta^{2}\cdot\nabla_{M}w=-\int_{M}(|\nabla_{M}w|^{2}\zeta^{2}+\zeta^{2}w\Delta_{M}w)$,
so if $w$ is a smooth solution of $\mathcal{L}_{M}w=0$ on $M\cap\breve{B}_{R}$
then

2.8
$\lambda\int_{M}|x|^{-2}w^{2}\zeta^{2}\,d\mu(x,y)\leqslant\int_{M}w^{2}|\nabla_{M}\zeta|^{2}\,d\mu,\quad\zeta\in
C^{1}_{c}(\breve{B}_{R}).$

We need one further preliminary for $M$, concerning asymptotics of $M$ at
$\infty$ in the case when $M$ is complete in all of $\varmathbb{R}^{n+1+\ell}$
with $M\subset U_{+}\times\varmathbb{R}^{\ell}$:

2.9 Lemma. If $M$ is a complete embedded minimal hypersurface in all of
$\varmathbb{R}^{n+1+\ell}$ satisfying 1.7 and 1.9, then $M$ has
$\varmathbb{C}$ with some constant integer multiplicity $q$ as its unique
tangent cone at $\infty$.

Furthermore if $M$ is stable (i.e.  _1.8_ holds with $\lambda=0$), then for
each $\delta>0$ there is $R_{0}=R_{0}(\varmathbb{C}_{0},q,\delta)>1$ such that

$None$
$M\setminus\bigl{(}B_{R_{0}}\cup\bigl{\\{}(x,y):|x|\leqslant\delta|y|\bigr{\\}}\bigr{)}\subset\cup_{j=1}^{q}{\rm
graph\,}u_{j}\subset M$

where each $u_{j}$ is a $C^{2}$ function on a domain $\Omega$ of
$\varmathbb{C}$ containing
$\varmathbb{C}\setminus(B_{R_{0}}\cup\bigl{\\{}(x,y):|x|\leqslant\delta|y|\bigr{\\}}$
and ${\rm
graph\,}u_{j}=\\{x+u_{j}(x,y)\nu_{\varmathbb{C}_{0}}(x):(x,y)\in\Omega\\}$
($\nu_{\varmathbb{C}_{0}}=$ unit normal of $\varmathbb{C}_{0}$ pointing into
$U_{+}$), and

$None$ $\lim_{R\to\infty}{\sup}_{(x,y)\in\Omega\setminus
B_{R}}\bigl{(}|x||\nabla_{\varmathbb{C}}^{2}u_{j}(x,y)|+|\nabla_{\varmathbb{C}}u_{j}(x,y)|+|x|^{-1}u_{j}(x,y)\bigr{)}=0$.

Proof: Let $C(M)$ be a tangent cone of $M$ at $\infty$. Thus $C(M)$ is a
stationary integer multiplicity varifold with $\lambda C(M)=C(M)$ for each
$\lambda>0$, and support of
$C(M)\subset\overline{U}_{+}\times\varmathbb{R}^{\ell}$ and, by the
compactness theorem of [SS81], $C(M)$ is stable and ${\rm sing\,\,}C(M)$ has
Hausdorff dimension $\leqslant n+\ell-7$. So by the maximum principle of
[Ilm96] and the constancy theorem, $C(M)=\varmathbb{C}$ with constant
multiplicity $q$ for some $q\in\\{1,2,\ldots\\}$.

Thus $\varmathbb{C}$, with multiplicity $q$, is the unique tangent cone of $M$
at $\infty$, and the “sheeting theorem” [SS81, Theorem 1] is applicable,
giving (i) and (ii). $\Box$

##  3\. Preliminaries concerning $\varmathbb{C}_{0}$ and $\varmathbb{C}$

$\varmathbb{C}_{0}\subset\varmathbb{R}^{n+1}\setminus\\{0\\}$ continues to
denote a smooth embedded minimal hypercone with ${\rm sing\,\,}\hskip
0.5pt\varmathbb{C}_{0}=\overline{\varmathbb}{C}_{0}\setminus\varmathbb{C}_{0}=\\{0\\}$,
$U_{\pm}$ denote the two connected components of
$\varmathbb{R}^{n+1}\setminus\overline{\varmathbb}{C}_{0}$, and we assume here
that $\varmathbb{C}_{0}$ is strictly stable as in 1.4.

With $\mathcal{L}_{\Sigma}$ the Jacobi operator of
$\Sigma=\varmathbb{C}_{0}\cap\varmathbb{S}^{n}$ as in 1.1, we let
$\varphi_{1}>0,\,\varphi_{2},\ldots$ be a complete orthonormal set of
eigenfunctions of $-L_{\Sigma}$. Thus

3.1 $\displaystyle-
L_{\Sigma}(\varphi_{j})=\lambda_{j}\varphi_{j},\,\,\,j=1,2,\ldots,\text{with
$\varphi_{1}>0$ and}$ $\displaystyle\hskip
79.49744pt\lambda_{1}<\lambda_{2}\leqslant\lambda_{3}\leqslant\cdots\leqslant\lambda_{j}\leqslant\cdots,\,\,\lambda_{j}\to\infty\text{
as }j\to\infty,$

and every $L^{2}(\Sigma)$ function $v$ can be written $v=\sum_{j}\langle
v,\varphi_{j}\rangle_{L^{2}(\Sigma)}\varphi_{j}$.

Notice that by orthogonality each $\varphi_{j},\,j\neq 1$, must then change
sign in $\Sigma$, and, with $\mathcal{L}_{\varmathbb{C}_{0}}$ the Jacobi
operator for $\varmathbb{C}_{0}$ as in 1.2,

3.2
$\mathcal{L}_{\varmathbb{C}_{0}}(r^{\gamma}\varphi_{j})=r^{\gamma-2}(\gamma^{2}+(n-2)\gamma-\lambda_{j})\,\varphi_{j},$

so in particular if $\varmathbb{C}_{0}$ is strictly stable (i.e. if
$\lambda_{1}>-({\textstyle\frac{n-2}{2}})^{2}$ as in 1.5) we have

3.3
$\displaystyle\mathcal{L}_{\varmathbb{C}_{0}}(r^{\gamma_{j}^{\pm}}\varphi_{j})=0,\quad\gamma_{j}^{\pm}=-{\textstyle\frac{n-2}{2}}\pm\bigl{(}({\textstyle\frac{n-2}{2}})^{2}+\lambda_{j}\bigr{)}^{1/2},$
$\displaystyle\hskip
20.0pt-\infty\leftarrow\gamma_{j}^{-}\leqslant\cdots\leqslant\gamma_{2}^{-}<\gamma_{1}^{-}<-{\textstyle\frac{n-2}{2}}<\gamma_{1}^{+}<\gamma_{2}^{+}\leqslant\cdots\leqslant\gamma_{j}^{+}\to\infty.$

Henceforth we write

3.4 $\gamma_{j}=\gamma_{j}^{+},\quad j=1,2,\ldots.$

The Jacobi operator $\mathcal{L}_{\varmathbb{C}}$ on the minimal cylinder
$\varmathbb{C}=\varmathbb{C}_{0}\times\varmathbb{R}^{\ell}$ is

3.5
$\mathcal{L}_{\varmathbb{C}}(v)=\mathcal{L}_{\varmathbb{C}_{0}}(v)+{\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}^{2}v,$

and we can decompose solutions $v$ of the equation
$\mathcal{L}_{\varmathbb{C}}v=0$ in terms of the eigenfunctions $\varphi_{j}$
of 3.1: for $v$ any given smooth function on $\varmathbb{C}\cap\breve{B}_{R}$,
we can write $v(r\omega,y)=\sum_{j}v_{j}(r,y)\varphi_{j}(\omega)$, where

3.6 $v_{j}(r,y)=\int_{\Sigma}v(r\omega,y)\,\varphi_{j}(\omega)\,d\mu(\omega),$

and then $\mathcal{L}_{\varmathbb{C}}v=0$ on $\breve{B}_{R}$ if and only if
$v_{j}$ satisfies

$r^{1-n}\frac{\partial}{\partial r}\bigl{(}r^{n-1}\frac{\partial
v_{j}}{\partial
r}\bigr{)}+{\textstyle\sum}_{k=1}^{\ell}\frac{\partial^{2}v_{j}}{\partial
y_{k}^{2}}-\frac{\lambda_{j}}{r^{2}}v_{j}=0$

for $(r,y)\in\breve{B}_{R}^{+}\,(=\\{(r,y):r>0,\,r^{2}+|y|^{2}<R^{2}\\})$.
Direct computation then shows that

3.7 $\frac{1}{r^{1+\beta}}\frac{\partial}{\partial
r}\Bigl{(}r^{1+\beta}\frac{\partial h}{\partial
r}\Bigr{)}+\sum_{k=1}^{\ell}\frac{\partial^{2}h}{\partial y_{k}^{2}}=0$

on $\breve{B}_{R}^{+}$, with

3.8
$h(r,y)=h_{j}(r,y)=r^{-\gamma_{j}}\int_{\Sigma}v(r\omega,y)\,\varphi_{j}(\omega)\,d\mu\text{
and }\beta=\beta_{j}=2\sqrt{({\textstyle\frac{n-2}{2}})^{2}+\lambda_{j}}.$

A solution $h$ of 3.7 (with any $\beta>0$) which has the property

3.9
$\int_{\\{(r,y):r^{2}+|y|^{2}<R^{2}\\}}r^{-2}h^{2}\,r^{1+\beta}\,drdy<\infty$

will be referred to as a _$\beta$ -harmonic function on_ $\\{(r,y):r\geqslant
0,\,r^{2}+|y|^{2}<R^{2}\\}$. Using 3.9 together with the weak form of the
equation 3.7 this is easily shown to imply the $W^{1,2}$ estimate

3.10
$\int_{\\{(r,y):r^{2}+|y|^{2}<\rho^{2}\\}}\bigl{(}(D_{r}h)^{2}+{\textstyle\sum}_{j=1}^{\ell}(D_{y_{j}}h)^{2}\bigr{)}\,r^{1+\beta}\,drdy<\infty$

for each $\rho<R$, and there is a unique such function with prescribed $C^{1}$
data $\varphi$ on $\bigl{\\{}(r,y):r^{2}+|y|^{2}=R^{2}\bigr{\\}}$, obtained,
for example, by minimizing $\int_{r^{2}+|y|^{2}\leqslant
R^{2}}(u_{r}^{2}+|u_{y}|^{2})\,r^{1+\beta}d\mu$ among functions with trace
$\varphi$ on $r^{2}+|y|^{2}=R^{2}$.

If $\beta$ is an integer (e.g. $\beta=1$ when $n=7$, $j=1$, and
$\varmathbb{C}_{0}$ is the “Simons cone”
$\bigl{\\{}x\in\varmathbb{R}^{8}:\sum_{i=1}^{4}(x^{i})^{2}=\sum_{i=5}^{8}(x^{i})^{2}\bigr{\\}}$)
then the $\beta\,$-Laplacian operator as in 3.7 is just the ordinary Laplacian
in $\varmathbb{R}^{2+\beta+\ell}$, at least as it applies to functions
$u=u(r,y)$ with $r=|x|$. Even for $\beta$ non-integer, there is, analogous to
the case when $\beta$ is an integer, $\beta$-harmonic polynomials of each
order (i.e. homogeneous polynomial solutions $h$ of 3.7) of each order
$q=0,1,\ldots$.

Indeed, as shown in the Appendix below (extending the discussion of [Sim94] to
arbitrary $\ell$ and at the same time showing the relevant power series
converge in the entire ball $\bigl{\\{}(r,y):r\geqslant
0,\,r^{2}+|y|^{2}<R^{2}\bigr{\\}}$ rather than merely in
$\bigl{\\{}(r,y):r\geqslant 0,\,r^{2}+|y|^{2}<\theta R^{2}\bigr{\\}}$ for
suitable $\theta\in(0,1)$), if $h=h(r,y)$ is a solution of the weak form of
3.7 on $\\{(r,y):r>0,\,r^{2}+|y|^{2}<R^{2}\\}$, i.e.

3.11
$\int_{\breve{B}_{1}^{+}}(u_{r}\zeta_{r}+u_{y}\cdot\zeta_{y})\,d\mu_{+}=0$

for all Lipschitz $\zeta$ with compact support in $r^{2}+|y|^{2}<R^{2}$ and if
$u$ satisfies 3.9, then $h$ is a real analytic function of the variables
$r^{2},y$, and, on all of $\\{(r,y):r\geqslant 0,\,r^{2}+|y|^{2}<R^{2}\\}$,
$h$ is a convergent sum

3.12 $h(r,y)={\textstyle\sum}_{q=0}^{\infty}h_{q}(r,y)\,\,\,\forall(r,y)\text{
with }r\geqslant 0\text{ and }\sqrt{r^{2}+|y|^{2}}<R,$

where $h_{q}$ is the degree $q$ homogeneous $\beta$-harmonic polynomial in the
variables $r^{2},y$ obtained by selecting the order $q$ terms in the power
series expansion of $h$.

In case $\ell=1$ there is, up to scaling, a unique $\beta$-harmonic polynomial
of degree $q$ of the form

$h_{q}=y^{q}+\sum_{k>0,\ell\geqslant 0,2k+j=q}c_{k\ell}r^{2k}y^{j},$

By direct computation, in case $\ell=1$ the $\beta$-harmonic polynomials
$h_{0},h_{1},h_{2},h_{3},h_{4}$ are respectively

$1,\,\,\,y,\,\,\,y^{2}-{\textstyle\frac{1}{2+\beta}}r^{2},\,\,\,y^{3}-{\textstyle\frac{3}{2+\beta}}r^{2}y,\,\,\,y^{4}-{\textstyle\frac{6}{2+\beta}}r^{2}y^{2}+{\textstyle\frac{3}{(2+\beta)(4+\beta)}}r^{4}.$

In the case $\ell\geqslant 2$, for each homogeneous degree $q$ polynomial
$p_{0}(y)$ there is a unique $\beta$-harmonic homogeneous polynomial $h$ of
degree $q$, with

$h(r,y)=\begin{cases}p_{0}(y)+{\textstyle\sum}_{j=1}^{q/2}r^{2j}p_{j}(y),&q\text{
even }\\\ p_{0}(y)+{\textstyle\sum}_{j=1}^{(q-1)/2}r^{2j}p_{j}(y),&q\text{
odd},\end{cases}$

where each $p_{j}(y)$ is a homogeneous degree $q-2j$ polynomial, and the
$p_{j}$ are defined inductively by
$p_{j+1}=-(2j+2)^{-1}(2j+2+\beta)^{-1}\Delta_{y}p_{j}$, $j\geqslant 0$.

In terms of spherical coordinates $\rho=\sqrt{r^{2}+|y|^{2}}$ and
$\omega=\rho^{-1}(r,y)\in\varmathbb{S}^{\ell}_{+}$,

$\varmathbb{S}^{\ell}_{+}=\\{\omega=(\omega_{1},\ldots,\omega_{\ell})\in\varmathbb{R}^{\ell+1}:\omega_{1}>0,\,|\omega|=1\\},$

the $\beta$-Laplacian $\Delta_{\beta}$ (i.e. the operator on the left of 3.7)
is

3.13
$\Delta_{\beta}h=\rho^{-\ell-1-\beta}\frac{\partial}{\partial\rho}\Bigl{(}\rho^{\ell+1+\beta}\frac{\partial
h}{\partial\rho}\Bigr{)}+\rho^{-2}\omega_{1}^{-1-\beta}{\rm
div}_{\varmathbb{S}_{+}^{\ell}}\bigl{(}\omega_{1}^{1+\beta}\nabla_{\varmathbb{S}_{+}^{\ell}}h\bigr{)}.$

Notice that in the case $\ell=1$ we can write $h=h(r,\theta)$ where
$\theta=\arctan(y/r)\in(-\pi/2,\pi/2)$ and 3.13 can be written

$\Delta_{\beta}h=\rho^{-2-\beta}\frac{\partial}{\partial\rho}\Bigl{(}\rho^{2+\beta}\frac{\partial
h}{\partial\rho}\Bigr{)}+\rho^{-2}\cos^{-1-\beta}\theta\frac{\partial}{\partial\theta}\bigl{(}\cos^{1+\beta}\theta\frac{\partial
h}{\partial\theta}\bigr{)}.$

Using 3.13 we see that the order $q$ homogeneous $\beta$-harmonic polynomials
$h_{q}$ (which are homogeneous of degree $q$ in the variable $\rho$) satisfy
the identity

3.14 ${\rm
div}_{\varmathbb{S}_{+}^{\ell}}\bigl{(}\omega_{1}^{1+\beta}\nabla_{\varmathbb{S}_{+}^{\ell}}h_{q}\bigr{)}=q(q+\ell+\beta)h_{q}\omega_{1}^{1+\beta}.$

Hence we have the orthogonality of $h_{p},h_{q}$ for $p\neq q$ on

3.15 $\varmathbb{S}^{\ell}_{+}=\\{(r,y):r>0,\,r^{2}+|y|^{2}=1\\}$

with respect to the measure $d\nu_{+}=\smash{{\omega_{1}}^{\\!1+\beta}}d\mu$
($\mu=\ell$-dimensional Hausdorff measure on $\varmathbb{S}_{+}^{\ell}$):

3.16 $\int_{\varmathbb{S}_{+}^{\ell}}h_{p}\,h_{q}\,d\nu_{+}=0\text{ for $p\neq
q$},\quad d\nu_{+}=\omega_{1}^{1+\beta}d\mu.$

Thus if $h$ satisfies 3.7 and 3.9 then

3.17 $\displaystyle\int_{B^{+}_{R}}h^{2}(r,y)\,r^{1+\beta}drdy$
$\displaystyle=\int_{0}^{R}\int_{\varmathbb{S}_{+}^{\ell}}h^{2}(\rho\omega)\,\rho^{1+\beta}\,d\nu_{+}\,\rho^{\ell}d\rho$
$\displaystyle=\sum_{q=0}^{\infty}(\ell+2+\beta+2q)^{-1}N_{q}^{2}R^{\ell+2+\beta+2q},$

where $B_{R}^{+}=\\{(r,y):r\geqslant 0,\,\,r^{2}+|y|^{2}\leqslant R\\}$ and
$N_{q}=\bigl{(}\int_{\varmathbb{S}_{+}^{\ell}}h_{q}^{2}(\omega)\,\omega_{1}^{1+\beta}d\mu\bigr{)}^{1/2}$
and $h_{q}(r,y)$ are as in 3.12.

Using 3.16 it is shown in the Appendix that the homogeneous $\beta$-harmonic
polynomials are complete in $L^{2}(\nu_{+})$, where
$d\nu_{+}=\omega_{1}^{1+\beta}d\mu_{\ell}$ on $\varmathbb{S}^{\ell}_{+}$. Thus
each $\varphi\in L^{2}(\nu_{+})$ can be written as an $L^{2}(\nu_{+})$
convergent series

3.18 $\varphi=\sum_{q=0}^{\infty}h_{q}|\varmathbb{S}^{\ell}_{+},$

with each $h_{q}$ either zero or a homogeneous degree $q$ $\beta$-harmonic
polynomial.

Next observe that if

3.19 $\mathcal{L}_{\varmathbb{C}}v=0\text{ on }\varmathbb{C}\cap B_{R}\text{
with }\int_{\varmathbb{C}\cap B_{R}}|x|^{-2}v^{2}\,d\mu<\infty,$

then each $h_{j}(r,y)$ defined as in 3.8 does satisfy the condition 3.9, and
hence we have

3.20
$v=\sum_{j=1}^{\infty}\sum_{q=0}^{\infty}r^{\gamma_{j}}h_{j,q}(r,y)\varphi_{j}(\omega),$

where $h_{j,q}$ is a homogeneous degree $q$ $\beta_{j}$-harmonic polynomial
and, using the orthogonality 3.16,

3.21 $\int_{\varmathbb{C}\cap
B_{\rho}}v^{2}\,d\mu=\rho^{\ell+2+\beta_{1}}\sum_{j=1}^{\infty}\sum_{q=0}^{\infty}(2+\ell+\beta_{j}+2q)^{-1}N_{j,q}^{2}\rho^{2q+\beta_{j}-\beta_{1}},\,\,\rho<R,$

where
$N_{j,q}=\bigl{(}\int_{\varmathbb{S}_{+}^{\ell}}h_{j,q}^{2}(\omega)\,d\nu_{+}\bigr{)}^{1/2}$.

Observe that (except for $(j,q)=(1,0)$) it is possible that
$r^{\gamma_{j}}h_{j,q}$ could have the same homogeneity as
$r^{\gamma_{i}}h_{i,p}$ (i.e. $\gamma_{j}+q=\gamma_{i}+p$) for some $i\neq
j,\,p\neq q$, but in any case, after some rearrangement of the terms in 3.21,
we see that there are exponents

$0=q_{1}<q_{2}<q_{2}<\cdots<q_{i}\to\infty,\quad
q_{i}=q_{i}(\varmathbb{C}_{0}),$

($q_{i}$ not necessarily integers, but nevertheless fixed depending only on
$\varmathbb{C}_{0}$) such that if $v$ is as in 3.19 then

3.22 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho}}v^{2}=\sum_{i=1}^{\infty}b_{i}^{2}\rho^{2q_{i}}$

(which in particular is an increasing function of $\rho$), $\rho\in(0,R]$, for
suitable constants $b_{j},\,j=1,2,\ldots$, where we use the notation

3.23 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho}}f=\rho^{-\ell-2-\beta_{1}}\int_{\varmathbb{C}\cap
B_{\rho}}f\,d\mu,\quad\rho>0.$

We claim that the logarithm of the right side of 3.22 is a convex function
$\psi(t)$ of $t=\log R$:

3.24 $\psi^{\prime\prime}(t)\geqslant 0\text{ where
}\psi(t)=\log\bigl{(}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho}}v^{2}\bigr{|}_{\rho=e^{t}}\bigr{)}\,\,\bigl{(}\,=\log({\textstyle\sum}_{i}b_{i}^{2}e^{2q_{i}t})\text{
by\leavevmode\nobreak\ \ref{l2-norm-v}}\bigr{)}.$

To check this, note that

$\psi^{\prime\prime}(t)=\psi^{-2}(t)\bigl{(}({\textstyle\sum}_{i}b_{i}^{2}e^{2q_{i}t})({\textstyle\sum}_{i}4q_{i}^{2}b_{i}^{2}e^{2q_{i}t})-({\textstyle\sum}_{i}2q_{i}b_{i}^{2}e^{2q_{i}t})^{2}\bigr{)}$

and by Cauchy-Schwarz this is non-negative for $t\in(0,R)$ and if there is a
$t_{0}\in(-\infty,\log R)$ such that $\psi^{\prime\prime}(t_{0})=0$ then there
is $i_{0}\in\\{1,2,\ldots\\}$ such that $b_{i}=0$ for every $i\neq i_{0}$, in
which case

3.25 $\psi(t)=\log b_{i_{0}}+2q_{i_{0}}t\quad t\in(-\infty,\log R).$

In particular the convexity 3.24 of $\psi$ implies

3.26 $\psi(t)-\psi(t-\log 2)\geqslant\psi(t-\log 2)-\psi(t-2\log 2),\quad
t\in(-\infty,\log R),$

and equality at any point $t\in(-\infty,\log R)$ implies 3.25 holds for some
$i_{0}$ and the common value of each side of 3.26 is $\log 4^{q_{i_{0}}}$.
Thus we see that if $Q\in(0,\infty)\setminus\\{4^{q_{1}},4^{q_{2}},\ldots\\}$
and $v$ is not identically zero then for each given $\rho\in(0,R]$

3.27 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho/2}}v^{2}\bigg{/}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho/4}}v^{2}\geqslant Q\Longrightarrow\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{\rho}}v^{2}\bigg{/}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap B_{\rho/2}}v^{2}>Q.$

3.28 Remark:  Notice that if $v_{1},\ldots,v_{q}$ are solutions of
$\mathcal{L}_{\varmathbb{C}}v=0$ on $\varmathbb{C}\cap B_{R}$ satisfying

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{R}}|x|^{-2}v_{j}^{2}(x,y)\,d\mu(x,y)<\infty\text{ for each }j=1,\ldots,q$

with $v_{j}\neq 0$ for some $j$, then
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{\varmathbb{C}\cap
B_{R}}\smash{\sum_{j=1}^{q}v_{j}^{2}\,d\mu}$ has again the form of  _3.22_ ,
so the implication  _3.27_ applies with the sum $\sum_{j=1}^{q}v_{j}^{\,2}$ in
place of $v^{2}$.

##  4\. Growth estimates for solutions of $\mathcal{L}_{\\!M}w=0$

Here we discuss growth estimates analogous to those used in [Sim83], [Sim85]
for solutions of $\mathcal{L}_{M}w=0$; in particular we discuss conditions on
$M$ and $w$ which ensure that the growth behaviour of solutions $w$ of
$\mathcal{L}_{M}w=0$ is analogous to that of the solutions $v$ of
$\mathcal{L}_{\varmathbb{C}}v=0$ discussed in the previous section.

The main growth lemma below (Lemma 4) applies locally in balls (so $M$ could
be a complete minimal hypersurface in a ball $\breve{B}_{R}$ rather than the
whole space), and we do not need the inclusion $M\subset
U_{+}\times\varmathbb{R}^{\ell}$. We in fact assume $R,\lambda,\Lambda>0$ and

4.1 $\left\\{\begin{aligned} &\text{\,$M\subset\breve{B}_{R}$ is embedded,
minimal, and satisfies\leavevmode\nobreak\ \ref{str-stab-M}}\,\,\text{ for
every }\,\zeta\in C^{1}_{c}(\breve{B}_{R}),\\\
\vskip-3.0pt\cr&\,R^{-n-\ell-2}\int_{M\cap
B_{R}}d^{2}(x)\,d\mu(x,y)<\varepsilon,\text{ and }R^{-n-\ell}\mu(M\cap
B_{R})\leqslant\Lambda,\end{aligned}\right.$

with an $\varepsilon$ (small) to be specified and with

4.2 $d(x)={\rm dist\,}(x,\varmathbb{C}_{0})\,\,(={\rm
dist\,}((x,y),\varmathbb{C}))\text{ for
}(x,y)\in\varmathbb{R}^{n+1}\times\varmathbb{R}^{\ell}.$

Taking $\zeta$ in 2.8 to be $1$ in $B_{\theta R}$, $0$ on $\partial B_{R}$,
and $|D\zeta|\leqslant 2/(1-\theta)$, we obtain

4.3 $\int_{M\cap B_{\theta
R}}\hskip-5.0pt|x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant CR^{-2}\int_{M\cap
B_{R}}\hskip-5.0ptw^{2}\,d\mu\,\,\forall\theta\in[{\textstyle\frac{1}{2}},1),\,\,C=C(\lambda,\theta).$

Notice that if we have a “doubling condition”

4.4 $\int_{M\cap B_{R}}w^{2}(x,y)\,d\mu(x,y)\leqslant K\int_{M\cap
B_{R/2}}w^{2}\,d\mu,$

then 4.3 implies that

$\int_{M\cap B_{\theta R}}\hskip-5.0pt|x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant
CKR^{-2}\int_{M\cap B_{R/2}}w^{2}\,d\mu,$

so, for $\delta^{-2}\geqslant 2CK$ (i.e. $\delta\leqslant(2CK)^{-1/2}$) we
have

4.5 $\int_{M\cap B_{\theta
R}}\hskip-5.0pt|x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant CR^{-2}\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|\leqslant\delta
R\\}}\hskip-5.0ptw^{2}\,d\mu\,\,\forall\theta\in[{\textstyle\frac{1}{2}},1),$

where $C=C(\lambda,\theta,K)$. The following lemma shows that in fact the
inequality 4.5 can be improved.

4.6 Lemma. For each $\lambda,\Lambda,K>0$ and
$\theta\in\smash{[{\textstyle\frac{1}{2}},1)}$ there is
$\varepsilon=\varepsilon(\lambda,\Lambda,\theta,K,\varmathbb{C})>0$ such that
if $M$ satisfies  _4.1_ and $\mathcal{L}_{M}w=0$ on $M\cap\breve{B}_{R}$, and
if the doubling condition  _4.4_ holds then

$\int_{M\cap B_{\theta R}}|x|^{-2}w^{2}(x,y)\,d\mu(x,y)\leqslant
CR^{-2}\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|\leqslant\frac{1}{3}R\\}}w^{2}\,d\mu,$

where $C=C(\lambda,\Lambda,\theta,K,\varmathbb{C})$.

Proof: Let $\theta\in[{\textstyle\frac{3}{4}},1)$. By scaling we can assume
$R=1$. If there is no $\varepsilon$ ensuring the claim of the lemma, then
there is a sequence $M_{k}$ of strictly stable hypersurfaces (with fixed
$\lambda$) with $\int_{M_{k}\cap B_{1}}d^{2}(x)\,d\mu(x,y)\to 0$ and a
sequence $w_{k}$ of solutions of $\mathcal{L}_{M_{k}}w_{k}=0$ such that

(1) $\int_{M_{k}\cap B_{1}}w_{k}^{2}(x,y)\,d\mu(x,y)\leqslant K\int_{M_{k}\cap
B_{1/2}}w_{k}^{2}\,d\mu,$

yet such that

(2) $\int_{M_{k}\cap B_{\theta}}|x|^{-2}w_{k}^{2}(x,y)\,d\mu(x,y)\geqslant
k\,\int_{M_{k}\cap
B_{1/2}\setminus\\{(x,y):|x|\leqslant\frac{1}{3}\\}}w_{k}^{2}\,d\mu.$

By 4.5,

(3) $\int_{M_{k}\cap B_{\theta}}|x|^{-2}w_{k}^{2}(x,y)\,d\mu(x,y)\leqslant
C\int_{M_{k}\cap
B_{1/2}\setminus\\{(x,y):|x|\leqslant\delta\\}}w_{k}^{2}\,d\mu,$

with $\delta=\delta(\lambda,\theta,K)$ and $C=C(\lambda,\theta,K)$.

Since $\int_{M_{k}\cap B_{1}}d^{2}\,d\mu_{k}\to 0$, in view of 2 (i), (ii)
there are sequences $\eta_{k}\downarrow 0$ and $\tau_{k}\uparrow 1$ with

(4) $\left\\{\begin{aligned}
&\,\,M_{k}\cap\breve{B}_{\tau_{k}}\setminus\\{(x,y):|x|\leqslant\eta_{k}\\}\subset\cup_{j=1}^{q}{\rm
graph\,}u_{k,j}\subset M_{k}\\\
&\qquad\sup(|u_{k,j}|+|\nabla_{\varmathbb{C}}u_{k,j}|+|\nabla^{2}_{\varmathbb{C}}u_{k,j}|)\to
0,\end{aligned}\right.$

where $q=q(\Lambda)\geqslant 1$ and each $u_{k,j}$ is $C^{2}$ on a domain
containing
$\varmathbb{C}\cap\breve{B}_{\tau_{k}}\setminus\\{(x,y):|x|\leqslant\eta_{k}\\}$.
Thus, with $\nu_{\varmathbb{C}_{0}}$ the unit normal of $\varmathbb{C}_{0}$
pointing into $U_{+}$ and with

(5) $w_{k,j}(x,y)=w_{k}((x+u_{k,j}(x,y)\nu_{\varmathbb{C}_{0}}(x),y)),$

for
$(x,y)\in\varmathbb{C}\cap\breve{B}_{\tau_{k}}\setminus\\{(x,y):|x|\leqslant\eta_{k}\\}$
and $j=1,\ldots,q$, we see that $w_{k,j}$ satisfies a uniformly elliptic
equation with coefficients converging in $C^{1}$ to the coefficients of
$\mathcal{L}_{\varmathbb{C}}$ on
$\Omega_{\sigma,\delta}=\varmathbb{C}\cap\breve{B}_{\sigma}\setminus\\{(x,y):|x|>\delta\\}$
for each $\sigma\in[{\textstyle\frac{1}{2}},1)$ and
$\delta\in(0,{\textstyle\frac{1}{2}}]$, hence, by Schauder estimates and (1),

(6) $|w_{k,j}|_{C^{2}(\Omega_{\sigma,\delta})}\leqslant
C\|w_{k}\|_{L^{2}(B_{1})}\leqslant CK\|w_{k}\|_{L^{2}(B_{1/2})}$

with $C$ independent of $k$, $k\geqslant k(\sigma,\delta)$. Hence, by (4) and
(6),

$\tilde{w}_{k,j}=({\textstyle\int}_{M_{k}\cap
B_{1/2}}w_{k}^{2}\,d\mu_{k})^{-1/2}w_{k,j}$

has a subsequence converging in $C^{1}$ locally in
$\varmathbb{C}\cap\breve{B}_{1}$ to a smooth solution $v_{j}$ of
$\mathcal{L}_{\varmathbb{C}}v_{j}=0$ on $\varmathbb{C}\cap\breve{B}_{1}$, and
since by (3) $\int_{M_{k}\cap
B_{1/2}\cap\\{(x,y):|x|\leqslant\sigma\\}}w_{k}^{2}\leqslant
C\sigma^{2}\int_{M_{k}\cap B_{1/2}}w_{k}^{2}$ for all
$\sigma\in(0,{\textstyle\frac{1}{2}}]$, we can then conclude

(7) $\int_{\varmathbb{C}\cap B_{1/2}}\sum_{j=1}^{q}v_{j}^{2}\,d\mu=1.$

But by (1) and (2)

$\int_{M_{k}\cap
B_{1/2}\setminus\\{(x,y):|x|\leqslant\frac{1}{3}\\}}w_{k}^{2}\,d\mu\leqslant
Ck^{-1}\int_{M_{k}\cap B_{1/2}}w_{k}^{2}\,d\mu,$

and so, multiplying each side by $({\textstyle\int}_{M_{k}\cap
B_{1/2}}w_{k}^{2}\,d\mu_{k})^{-1/2}$ and taking limits, we conclude

$\int_{\varmathbb{C}\cap
B_{1/2}\setminus\\{(x,y):|x|\leqslant\frac{1}{3}\\}}\sum_{j=1}^{p}v_{j}^{2}=0.$

In view of (7) this contradicts unique continuation for solutions of
$\mathcal{L}_{\varmathbb{C}}v=0$ (applicable since the solutions $v$ of
$\mathcal{L}_{\varmathbb{C}}v=0$ are real-analytic). $\Box$

We can now establish the growth lemma.

4.7 Lemma. For each $\lambda,\Lambda>0$,
$Q\in(0,\infty)\setminus\\{4^{q_{1}},4^{q_{2}},\ldots\\}$
($q_{1},q_{2},\ldots$ as in 3.22), and $\alpha\in[{\textstyle\frac{1}{2}},1)$,
there is $\varepsilon=\varepsilon(Q,\alpha,\lambda,\Lambda,\varmathbb{C})>0$
such that if $M$ satisfies  _4.1_ and if $\mathcal{L}_{M}w=0$ on
$M\cap\breve{B}_{R}$ then

$None$ $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}}w^{2}\geqslant Q\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/4}}w^{2}\Longrightarrow\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{B_{R}}w^{2}\geqslant Q\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{B_{R/2}}w^{2},$

and

$None$ $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}w^{2}\geqslant\alpha\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}}w^{2},$

where we use the notation (analogous to 3.23)

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{\rho}}f=\rho^{-\ell-2-\beta}\int_{M\cap B_{\rho}}f\,d\mu.$

Notice that no hypothesis like $M\subset U_{+}\times\varmathbb{R}^{\ell}$ is
needed here.

Proof: By scaling we can assume $R=1$. If there is
$Q\in(0,\infty)\setminus\\{4^{q_{1}},4^{q_{2}},\ldots\\}$ such that there is
no $\varepsilon$ ensuring the first claim of the lemma, then there is a
sequence $M_{k}$ satisfying 1.8 (with fixed $\lambda$) with $\int_{M_{k}\cap
B_{1}}d^{2}(x)\,d\mu(x,y)\to 0$ and a sequence $w_{k}$ of solutions of
$\mathcal{L}_{M_{k}}w_{k}=0$, such that

(1) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap
B_{1/2}}w_{k}^{2}\geqslant Q\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/4}}w_{k}^{2}\text{ and
}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap
B_{1}}w_{k}^{2}<Q\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap
B_{1/2}}w_{k}^{2}.$

The latter inequality implies we have the doubling condition 4.4 with $K=Q$,
so we can repeat the compactness argument in the proof Lemma 4. Thus by (1),
with the same notation as in the proof of 4, we get convergence of $w_{k,j}$
to a smooth solution $v_{j}$ of $\mathcal{L}_{\varmathbb{C}}v_{j}=0$ on
$\varmathbb{C}\cap\breve{B}_{1}$ with
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{\varmathbb{C}\cap
B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu=1$, $\int_{\varmathbb{C}\cap
B_{1}}|x|^{-2}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}<\infty$,

(2) $\displaystyle\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\geqslant Q\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1/4}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\text{ and }$
$\displaystyle\hskip 72.26999pt\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\leqslant Q\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu.$

In view of 3, this contradicts 3.27.

Similarly, if the second claim of the lemma fails for some
$\alpha\in[{\textstyle\frac{1}{2}},1)$, after taking a subsequence of $k$
(still denoted $k$) we get sequences $M_{k},w_{k}$ and $w_{k,j}$ with

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap
B_{1}}w_{k}^{2}\,d\mu<\alpha\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap B_{1/2}}w_{k}^{2}\,d\mu$

(i.e. the doubling condition 4.4 with $K=\alpha$), and smooth solutions
$v_{j}=\lim w_{k,j}$ of $\mathcal{L}_{\varmathbb{C}}v_{j}=0$ on
$\varmathbb{C}\cap\breve{B}_{1}$ with $0<\int_{M\cap
B_{1}}|x|^{-2}v_{j}^{2}<\infty$ and

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu\leqslant\alpha\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1/2}}{\textstyle\sum}_{j=1}^{q}v_{j}^{2}\,d\mu,$

which is impossible by 3.22 and 3. $\Box$

Since $\mathcal{L}_{M}$ is the linearization of the minimal surface operator
$\mathcal{M}_{M}$, a smooth family $\\{M_{t}\\}_{|t|<\varepsilon}$ of minimal
submanifolds with $M_{0}=M$ must generate a velocity vector $v$ at $t=0$ with
normal part $w=v\cdot\nu_{M}$ ($\nu_{M}$ a smooth unit normal of $M$) being a
solution of the equation $\mathcal{L}_{M}w=0$. In particular the family of
homotheties $(1+t)M$ generates the solution $w=(x,y)\cdot\nu_{M}(x,y)$ and the
translates $M+te_{n+1+j}$ generate the solutions
$w=e_{n+1+j}\cdot\nu_{M}(x,y)=\nu_{y_{j}}(x,y)$, $j=1,\ldots,\ell$. Thus

4.8 $w=(x,y)\cdot\nu_{M}\text{ and }w=\nu_{y_{j}}\text{ both satisfy
}\mathcal{L}_{M}w=0\text{ and the inequality }\ref{pre-gro-1}.$

4.9 Corollary. Suppose $\Lambda,\lambda,\gamma>0$, $q\in\\{1,2,\ldots\\}$, and
assume $M$ is a complete, embedded, minimal hypersurface in
$\varmathbb{R}^{n+1+\ell}$, strict stability 1.8 holds, and $M$ has
$\varmathbb{C}$ with some multiplicity $q$ as its unique tangent cone at
$\infty$. If $\mathcal{L}_{M}w=0$, and
$\sup_{R>1}R^{-\gamma}\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M\cap
B_{R}}w^{2}<\Lambda$, then there is $R_{0}=R_{0}(M,\gamma,\lambda,\Lambda,q)$
such that for all $R\geqslant R_{0}$ we have the “strong doubling condition”

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}|x|^{-2}w^{2}\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}w^{2},\quad
C=C(\gamma,q,\lambda,\Lambda).$

4.10 Remarks: (1) Observe that in case $M\subset
U_{+}\times\varmathbb{R}^{\ell}$ the unique tangent cone assumption here is
automatically satisfied by virtue of Lemma 2.

(2) Since $\mu(M\cap B_{R})\leqslant q\mu(\varmathbb{C}\cap B_{1})R^{n+\ell}$,
$|(x,y)\cdot\nu_{M}(x,y)|\leqslant R$, and $|\nu_{y_{j}}|\leqslant 1$ on
$M\cap B_{R}$, in view of 4.8 the above lemma applies to both
$w=(x,y)\cdot\nu_{M}$ and $w=\nu_{y_{j}}$ with $\gamma=2|\gamma_{1}|+2$ and
$\gamma=2|\gamma_{1}|$ respectively.

Proof of Corollary 4: We can of course assume that $w$ is not identically
zero. In view of Lemma 2, for any $\tilde{\gamma}>\gamma$ with
$2^{\tilde{\gamma}}\in(1,\infty)\setminus\\{4^{q_{1}},4^{q_{2}},\ldots\\}$,
$M\cap\breve{B}_{R}$ satisfies the hypotheses of Lemma 4 with
$Q=2^{\tilde{\gamma}}$ for all sufficiently large $R$, so

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{2^{k}R}}w^{2}\geqslant 2^{\tilde{\gamma}}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{2^{k-1}R}}w^{2}\Longrightarrow\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B_{2^{k+1}R}}w^{2}\geqslant
2^{\tilde{\gamma}}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{2^{k}R}}w^{2}$

for $k=1,2,\ldots$ and for any choice of $R$ sufficiently large (depending on
$M,\tilde{\gamma},\lambda$), and hence, by iteration, if
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M\cap
B_{R}}w^{2}\geqslant
2^{\tilde{\gamma}}\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M\cap
B_{R/2}}w^{2}$ we would have
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M\cap
B_{2^{k}R}}w^{2}\geqslant C2^{k\tilde{\gamma}}$, $k=1,2,\ldots$ with $C>0$
independent of $k$ , contrary to the hypothesis on $w$ since
$C2^{k(\tilde{\gamma}-\gamma)}>\Lambda$ for sufficiently large $k$. Thus, with
such a choice of $\tilde{\gamma}$, we have the doubling condition 4.4 with
$K=2^{\tilde{\gamma}}$:

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B_{R}}w^{2}\leqslant
2^{\tilde{\gamma}}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}}w^{2}$

for all $R\geqslant R_{0}$, $R_{0}=R_{0}(M,\tilde{\gamma})$. Then the required
result is proved by Lemma 4. $\Box$

##  5\. Asymptotic $L^{2}$ Estimates for $d$

The following lemma gives an $L^{2}$ estimate for the distance function
$d|M\cap B_{R}$ as $R\to\infty$ in case $M$ is as in Theorem 1, where as usual

$d(x)={\rm dist\,}(x,\varmathbb{C}_{0})\,\,(={\rm
dist\,}((x,y),\varmathbb{C})\text{ for
}(x,y)\in\varmathbb{R}^{n+1}\times\varmathbb{R}^{\ell}).$

In this section we continue to use the notation

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{\rho}}f=\rho^{-\ell-2-\beta_{1}}\int_{M\cap B_{\rho}}f\,d\mu.$

5.1 Lemma. Let $\alpha,\lambda,\Lambda>0$ and assume $M$ is embedded, minimal,
$M\subset U_{+}\times\varmathbb{R}^{\ell}$, $R^{-n-\ell}\mu(M\cap
B_{R})\leqslant\Lambda$ for each $R>1$ and $M$ satisfies the strict stability
1.8. Then there is $R_{0}=R_{0}(M,\lambda,\Lambda,\alpha)>1$ such that

$C^{-1}R^{-\alpha}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R_{0}}}d^{2}\leqslant\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}d^{2}\leqslant CR^{\alpha}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B_{R_{0}}}d^{2}\,\,\,\,\forall
R\geqslant R_{0},\quad C=C(\lambda,\Lambda,\varmathbb{C}_{0}).$

To facilitate the proof we need the technical preliminaries of the following
lemma. In this lemma

5.2 $\nu_{M}=(\nu_{1},\ldots,\nu_{n+\ell+1})$

continues to denote a smooth unit normal for $M$, $\nu_{\varmathbb{C}_{0}}$
continues to denote the unit normal of $\varmathbb{C}_{0}$ pointing into
$U_{+}$, and
$\varepsilon_{0}=\varepsilon_{0}(\varmathbb{C}_{0})\in(0,{\textstyle\frac{1}{2}}]$
is assumed small enough to ensure that there is a smooth nearest point
projection

5.3
$\pi:\bigl{\\{}x\in\varmathbb{R}^{n+1}:d(x)<\varepsilon_{0}|x|\bigr{\\}}\to\varmathbb{C}_{0}.$

5.4 Lemma. Suppose $\Lambda>0$, $\delta\in(0,{\textstyle\frac{1}{2}}]$,
$\theta\in[{\textstyle\frac{1}{2}},1)$, $M$ satisfies 4.1, and $M\subset
U_{+}\times\varmathbb{R}^{\ell}$. Then there is
$\varepsilon=\varepsilon(\delta,\lambda,\Lambda,\theta,\varmathbb{C})\in(0,\varepsilon_{0}]$
such that, for all $(x,y)\in M\cap B_{\theta R}$ with $d(x)<\varepsilon|x|$,

$None$
$\min\bigl{\\{}\bigl{|}\nu_{M}(x,y)-\bigl{(}\nu_{\varmathbb{C}_{0}}(\pi(x)),0\bigr{)}\bigr{|},\,\bigl{|}\nu_{M}(x,y)+\bigl{(}\nu_{\varmathbb{C}_{0}}(\pi(x)),0\bigr{)}\bigr{|}\bigr{\\}}<\delta$

in particular
$|\nu_{y}|(x,y)\,\,(\,=|(\nu_{y_{1}}(x,y),\ldots,\nu_{y_{\ell}}(x,y))|)<\delta$),
and

(ii) $\displaystyle(1-7\delta)|x|^{-1}|(x,0)\cdot\nu_{M}(x,y)|\leqslant$
$\displaystyle\hskip
72.26999pt\bigl{|}(x,0)\cdot\nabla_{M}\bigl{(}d(x)/|x|\bigr{)}\bigr{|}\leqslant(1+7\delta)|x|^{-1}|(x,0)\cdot\nu_{M}(x,y)|.$

Remark: The inequalities in (ii) do not depend on the minimality of $M$—(ii)
is true for any smooth hypersurface $M$ at points where $d(x)<\varepsilon|x|$
and where (i) holds for suitably small $\delta\in(0,{\textstyle\frac{1}{2}}]$.

Proof of Lemma 5: By scaling we can assume $R=1$. If (i) fails for some
$\Lambda,\delta,\theta$ then there is a sequence $(\xi_{k},\eta_{k})\in M\cap
B_{\theta}$ with $d(\xi_{k})\leqslant k^{-1}|\xi_{k}|$ and

(1)
$\min\bigl{\\{}|\nu(\xi_{k},\eta_{k})-\nu_{\varmathbb{C}_{0}}(\pi(\xi_{k}))|,|\nu(\xi_{k},\eta_{k})+\nu_{\varmathbb{C}_{0}}(\pi(\xi_{k}))|\bigr{\\}}\geqslant\delta.$

Then passing to a subsequence we have
$\tilde{\xi}_{k}=|\xi_{k}|^{-1}\xi_{k}\to\xi\in\varmathbb{C}_{0}$ and
$M_{k}=|\xi_{k}|^{-1}(M-(0,\eta_{k}))$ converges in the varifold sense to an
integer multiplicity $V$ which is stationary in $B_{\theta^{-1}}(\xi,0)$,
${\rm spt\,}V\subset\overline{U}_{+}\times\varmathbb{R}^{\ell}$, and
$\xi\in{\rm spt\,}V\cap\varmathbb{C}_{0}$. Also, by the compactness and
regularity theory of [SS81], ${\rm sing\,\,}V\cap\breve{B}_{\theta^{-1}}$ has
Hausdorff dimension $\leqslant n+\ell-7$. So, since $\varmathbb{C}_{0}$ is
connected, by the maximum principle of [Ilm96] we have

(2) $V\hbox{ {\vrule height=6.25963pt}{\leaders\hrule\hskip 5.69046pt}
}\breve{B}_{\theta^{-1}}(\xi,0)=(q\varmathbb{C}+W)\hbox{ {\vrule
height=6.25963pt}{\leaders\hrule\hskip 5.69046pt}
}\breve{B}_{\theta^{-1}}(\xi,0)$

where $W$ is integer multiplicity, stationary in $\breve{B}_{\theta^{-1}}$ and
with ${\rm spt\,}W\subset\overline{U}_{+}\times\varmathbb{R}^{\ell}$. Taking
the maximum integer $q$ such that this holds, we then have

(3) $B_{\sigma}(\xi,0)\cap{\rm spt\,}W=\emptyset$

for some $\sigma>0$, because otherwise ${\rm
spt\,}W\cap\overline{\varmathbb}{C}\neq\emptyset$ and we could apply [Ilm96]
again to conclude $W\hbox{ {\vrule height=6.25963pt}{\leaders\hrule\hskip
5.69046pt} }\breve{B}_{\theta^{-1}}(\xi,0)=(\varmathbb{C}+W_{1})\hbox{ {\vrule
height=6.25963pt}{\leaders\hrule\hskip 5.69046pt}
}\breve{B}_{\theta^{-1}}(\xi,0)$ where $W_{1}$ is stationary integer
multiplicity, contradicting the maximality of $q$ in (2).

In view of (2), (3) we can then apply the sheeting theorem [SS81, Theorem 1]
to assert that $M_{k}\cap B_{\sigma/2}(\xi,0)$ is $C^{2}$ close to
$\varmathbb{C}$, and in particular

$\min\bigl{\\{}|\nu(\xi_{k},\eta_{k})-\nu_{\varmathbb{C}_{0}}(\pi(\xi_{k}))|,|\nu(\xi_{k},\eta_{k})+\nu_{\varmathbb{C}_{0}}(\pi(\xi_{k}))|\bigr{\\}}\to
0,$

contradicting (1).

To prove (ii), let $(x_{0},y_{0})\in M$ such that
$d(x_{0})<\varepsilon|x_{0}|$ and such that (i) holds with
$(x,y)=(x_{0},y_{0})$, and let $\sigma>0$ be small enough to ensure that both
$d(x)<\varepsilon|x|$ and (i) hold for all $(x,y)\in M\cap
B_{\sigma}(x_{0},y_{0})$. Let

$M_{y_{0}}=\bigl{\\{}x\in\varmathbb{R}^{n+1}:(x,y_{0})\in M\cap
B_{\sigma}(x_{0},y_{0})\bigr{\\}}.$

Then, taking a smaller $\sigma$ if necessary, we can assume

(4) $M_{y_{0}}={\rm
graph\,}u=\bigl{\\{}(\xi+u(\xi)\nu_{\varmathbb{C}_{0}}(\xi),y_{0}):\xi\in\Omega\bigr{\\}},$

where $\nu_{\varmathbb{C}_{0}}$ is the unit normal of $\varmathbb{C}_{0}$
pointing into $U_{+}$, $\Omega$ is an open neighborhood of $\pi(x_{0})$ in
$\varmathbb{C}_{0}$, and $u$ is $C^{2}(\Omega)$.

Then with $x=\xi+u(\xi)\nu_{\varmathbb{C}_{0}}(\xi)\in M_{y_{0}}$ we have also
$x(t)=t\xi+u(t\xi)\nu_{\varmathbb{C}_{0}}(\xi)\in M_{y_{0}}$ for $t$
sufficiently close to $1$ (because
$\nu_{\varmathbb{C}_{0}}(t\xi)=\nu_{\varmathbb{C}_{0}}(\xi)$) and hence

(5)
${\textstyle\frac{d}{dt}}\bigr{|}_{t=1}x(t)=\xi+\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)\nu_{\varmathbb{C}_{0}}(\xi)\in
T_{x}M_{y_{0}}.$

With $\nu^{\prime}(x)=(\nu_{1}(x,y_{0}),\ldots,\nu_{n+1}(x,y_{0}))$ (which is
normal to $M_{y_{0}}$), we can assume, after changing the sign of $\nu$ if
necessary, that

(6) $\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x)>0\text{ for }x\in
M_{y_{0}},\text{ so
}|\nu_{\varmathbb{C}_{0}}(\xi)-\nu^{\prime}(x)|<\delta\text{ by (i)}.$

By (5) we have

$x\cdot\nu^{\prime}(x)=\bigl{(}x-{\textstyle\frac{d}{dt}}\bigr{|}_{t=1}x(t)\bigr{)}\cdot\nu^{\prime}(x)=\bigl{(}u(\xi)-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)\bigr{)}\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x),$

hence

(7)
$u(\xi)-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)=(\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x))^{-1}x\cdot\nu^{\prime}(x).$

Differentiating the identity
$u(t\xi)=d\bigl{(}t\xi+u(t\xi)\nu_{\varmathbb{C}_{0}}(\xi)\bigr{)}$ at $t=1$,
we obtain

$\displaystyle-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)$
$\displaystyle=-\bigl{(}\xi+\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)\nu_{\varmathbb{C}_{0}}(\xi)\bigr{)}\cdot\nabla_{M_{y_{0}}}d(x)$
$\displaystyle=-\bigl{(}x-(u(\xi)-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi))\nu_{\varmathbb{C}_{0}}(\xi)\bigr{)}\cdot\nabla_{M_{y_{0}}}d(x)$
$\displaystyle=-x\cdot\nabla_{M_{y_{0}}}d(x)+\bigl{(}(u(\xi)-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi))\nu_{\varmathbb{C}_{0}}(\xi)\bigr{)}\cdot\nabla_{M_{y_{0}}}d(x),$

hence, adding $d(x)\,\,(\,=u(\xi))$ to each side of this identity,

(8) $\displaystyle d(x)-x\cdot\nabla_{M_{y_{0}}}d(x)$
$\displaystyle=\bigl{(}u-\xi\cdot\nabla_{\varmathbb{C}_{0}}u(\xi)\bigr{)}\bigl{(}1-\nu_{\varmathbb{C}_{0}}(\xi).\nabla_{M_{y_{0}}}d(x)\bigr{)}$
$\displaystyle=\bigl{(}\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x)\bigr{)}^{-1}x\cdot\nu^{\prime}(x)\bigl{(}1-\nu_{\varmathbb{C}_{0}}(\xi).\nabla_{M_{y_{0}}}d(x)\bigr{)}$

by (7). Using the identity
$|x|^{-1}x\cdot\nabla_{M_{y_{0}}}|x|=1-|\nu^{\prime}(x)|^{-2}(x\cdot\nu^{\prime}(x)/|x|)^{2}$
we see that the left side of (8) is
$\bigl{(}-x\cdot\nabla_{M_{y_{0}}}(d(x)/|x|)-|\nu^{\prime}(x)|^{-2}(|x\cdot\nu^{\prime}(x)|/|x|)^{2}\bigr{)}|x|$,
so (8) gives

$x\cdot\nabla_{M_{y_{0}}}(d(x)/|x|)=|x|^{-1}x\cdot\nu^{\prime}(x)(1+E),$

where

$E=|\nu^{\prime}(x)|^{-2}x\cdot\nu^{\prime}(x)/|x|+(|\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x)|^{-1}-1)-|\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nu^{\prime}(x)|^{-1}\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nabla_{M_{y_{0}}}d(x).$

Since
$\nu_{\varmathbb{C}_{0}}(\xi)\cdot\nabla_{M_{y_{0}}}d(x)=(\nu_{\varmathbb{C}_{0}}(\xi)-\nu^{\prime}(x))\cdot\nabla_{M_{y_{0}}}d(x)$
and
$x\cdot\nu^{\prime}(x)=x\cdot\nu_{\varmathbb{C}_{0}}(\xi)+x\cdot(\nu^{\prime}(x)-\nu_{\varmathbb{C}_{0}}(\xi))=d(x)-x\cdot(\nu_{\varmathbb{C}_{0}}(\xi)-\nu^{\prime}(x))$,
and $|\nu_{\varmathbb{C}_{0}}(\xi)-\nu^{\prime}(x)|<\delta$ by (6), direct
calculation then shows that $|E|<7\delta$. Since
$x\cdot\nabla_{M_{y_{0}}}(d(x)/|x|)=(x,0)\cdot\nabla_{M}(d(x)/|x|)\bigr{|}_{(x,y_{0})}$
and $x\cdot\nu^{\prime}(x)=(x,0)\cdot\nu(x,y_{0})$ we thus have the required
inequalities (ii) at $(x_{0},y_{0})$. $\Box$

Proof of Lemma 5. First we establish an $L^{2}$ bound for $|x|^{-1}d(x)$,
namely

(1) $\int_{M\cap B_{R}}|x|^{-2}d^{2}(x,y)\,d\mu(x,y)\leqslant
CR^{-2}\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}d^{2}(x,y)\,d\mu(x,y)$

for $R$ sufficiently large (depending on $M$), where
$C=C(\lambda,\Lambda,\varmathbb{C})$.

To check this observe that since $R^{-n-\ell-2}\int_{M\cap B_{R}}d^{2}\to 0$
by 2, we can apply Lemma 5 for sufficiently large $R$ giving conclusions
(i),(ii) of 5 with $\delta={\textstyle\frac{1}{8}}$ and
$\theta={\textstyle\frac{1}{2}}$.

Let
$\varepsilon=\varepsilon(\delta,\lambda,\Lambda,\theta)\in(0,\varepsilon_{0}]$
be as in Lemma 5 with $\delta={\textstyle\frac{1}{8}}$ and
$\theta={\textstyle\frac{1}{2}}$, and let
$\bigl{(}d/|x|\bigr{)}_{\varepsilon}=\min\\{d(x)/|x|,\varepsilon\\}$. Then,
since $d(x)/|x|\leqslant 1$ for all $x$,

(2)
$d(x)/|x|\leqslant\varepsilon^{-1}\bigl{(}d(x)/|x|\bigr{)}_{\varepsilon}\text{
at all points of }M,$

and since $\nabla_{M}\bigl{(}d(x)/|x|\bigr{)}_{\varepsilon}=0$ for $\mu$-a.e.
point $(x,y)$ with $d(x)/|x|\geqslant\varepsilon$, by 5(ii)

(3) $|(x,0)\cdot\nabla_{M}\bigl{(}d(x)/|x|\bigr{)}_{\varepsilon}|\leqslant
2|x|^{-1}|(x,0)\cdot\nu(x,y)|\text{ at $\mu$-a.e.\ point of }M\setminus
B_{R_{0}},$

for suitable $R_{0}=R_{0}(M,\lambda,\Lambda,\varmathbb{C})$.

Take $R>R_{0}$ and $\zeta\in C^{1}_{c}(\breve{B}^{n+1}_{R})$, $\chi\in
C^{1}_{c}(\breve{B}^{\ell}_{R})$ with $\zeta(x)\equiv 1$ for
$|x|\leqslant{\textstyle\frac{1}{4}}R$, $\zeta(x)=0$ for
$|x|>{\textstyle\frac{1}{2}}R$, $\chi(y)\equiv 1$ for
$|y|\leqslant{\textstyle\frac{1}{4}}R$, $\chi(y)=0$ for
$|y|>{\textstyle\frac{1}{2}}R$, and $|D\zeta|,|D\chi|\leqslant 3R^{-1}$. Since

${\rm div}_{M}(x,0)\geqslant n>2\text{ and
}|(x,0)\cdot\nabla_{M}\chi(y)|\leqslant|x|\,|D\chi(y)|\,|\nu_{y}|,$

we can apply (2), (3), and 2.1 with
$Z_{|(x,y)}=\bigl{(}d(x)/|x|\bigr{)}_{\varepsilon}^{2}\zeta^{2}(x)\chi^{2}(y)(x,0)$
to conclude

(4) $\displaystyle\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/4}}\hskip-8.0pt|x|^{-2}d^{2}(x)\,d\mu$ $\displaystyle\hskip
14.45377pt\leqslant C\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}\hskip-6.0pt\bigl{(}|x|^{-2}((x,0)\cdot\nu)^{2}+|\nu_{y}|^{2}\bigr{)}\,d\mu+CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}\setminus\\{(x,y):|x|<\frac{1}{4}R\\}}\hskip-20.0ptd^{2}\,d\mu$
$\displaystyle\hskip 14.45377pt\leqslant 2C\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}\hskip-6.0pt\bigl{(}|x|^{-2}((x,y)\cdot\nu)^{2}+R^{2}|x|^{-2}|\nu_{y}|^{2}\bigr{)}\,d\mu+$
$\displaystyle\hskip 187.90244ptCR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}\setminus\\{(x,y):|x|<\frac{1}{4}R\\}}\hskip-20.0ptd^{2}\,d\mu,$

where $C=C(\lambda,\Lambda,\varmathbb{C})$.

By 4, with either of the choices $w=(x,y)\cdot\nu$ or $w=R\nu_{y_{j}}$,

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}|x|^{-2}w^{2}\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}w^{2},$

and hence (4) gives

(5) $\displaystyle\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/4}}\hskip-8.0pt|x|^{-2}d^{2}(x)\,d\mu\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}\hskip-20.0pt(\,((x,y)\cdot\nu)^{2}+R^{2}|\nu_{y}|^{2}\,)$
$\displaystyle\hskip 180.67499pt+CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}\hskip-30.0ptd^{2}\,d\mu.$

In view of 2 there exists $R_{0}>1$ and
$\delta,\tilde{\delta}:(0,\infty)\to(0,\infty)$ such that
$\delta(t),\tilde{\delta}(t)\to 0$ as $t\to\infty$ and

(6) $\left\\{\begin{aligned} &\bigl{\\{}(x,y)\in
M:|x|\leqslant\tilde{\delta}(|(x,y))|(x,y)|\bigr{\\}}\setminus
B_{R_{0}}\subset\cup_{j=1}^{q}{\rm graph\,}u_{j}\subset M\\\
&{\sup}_{(\xi,\eta)\in\Omega\setminus
B_{R}}{\textstyle\sum}_{j=1}^{q}\bigl{(}|Du_{j}(\xi,\eta)|+|(\xi,\eta)|^{-1}u_{j}(\xi,\eta)\bigr{)}\to
0\text{ as }R\to\infty,\end{aligned}\right.$

where $u_{j}$ are positive $C^{2}$ functions on the domain $\Omega$,

$\Omega\supset\\{(x,y)\in\varmathbb{C}:|x|<\delta((|x,y|))|(x,y)|\\}\setminus
B_{R_{0}}.$

For $(x,y)=(\xi+u_{j}(\xi,y)\nu_{\varmathbb{C}_{0}}(\xi),y)$ with
$(\xi,y)\in\Omega$, take an orthonormal basis $\tau_{1},\ldots,\tau_{n+\ell}$
for $T_{(\xi,y)}\varmathbb{C}$ with $\tau_{1},\ldots,\tau_{n-1}$ principal
directions for $\varmathbb{C}_{0}\cap\varmathbb{S}^{n}$ (so
$\nabla_{\tau_{i}}\nu_{\varmathbb{C}_{0}}=\kappa_{i}\tau_{i}$ for
$i=1,\ldots,n-1$), $\tau_{n}=|\xi|^{-1}\xi$, and $\tau_{n+j}=e_{n+1+j}$,
$j=1,\ldots,\ell$. Then the unit normal $\nu(x,y)$ of $M$ is

$\displaystyle\nu(x,y)=\bigl{(}1+{\textstyle\sum}_{i=1}^{n}(1+\kappa_{i}u_{j}(\xi,y))^{-2}(D_{\tau_{i}}u_{j}(\xi,y))^{2}+|D_{y}u(\xi,y)|^{2}\bigr{)}^{-1/2}$
$\displaystyle\hskip
14.45377pt\times\bigl{(}\nu_{\varmathbb{C}_{0}}(\xi,y)-{\textstyle\sum}_{i=1}^{n}(1+\kappa_{i}u_{j}(\xi,y))^{-1}D_{\tau_{i}}u_{j}(\xi,y)\tau_{i}-{\textstyle\sum}_{k=1}^{\ell}D_{y_{k}}u_{j}(\xi,y)e_{n+1+k}\bigr{)},$

so for $R$ sufficiently large

(7) $|\nu_{y}(x,y)|\leqslant|D_{y}u_{j}(\xi,y)|\text{ and
}|(x,0)\cdot\nu_{M}(x,y)|\leqslant|u_{j}(\xi,y)|+2|\xi||\nabla_{\varmathbb{C}}u_{j}(\xi,y)|.$

Also, since $u_{j}$ satisfies the equation
$\mathcal{M}_{\varmathbb{C}}(u_{j})=0$, where $\mathcal{M}_{\varmathbb{C}}$
satisfies 2.7 with $\varmathbb{C}$ in place of $M$, we have

(8)
$\int_{B_{R}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}|\nabla_{\varmathbb{C}}u_{j}|^{2}\leqslant
CR^{-2}\int_{B_{3R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}u_{j}^{2},$

Also, $d(\xi+u_{j}(\xi,y)\nu_{\varmathbb{C}_{0}}(\xi))=u_{j}(\xi,y)$ so, by
(7) and (8),

(9) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}w^{2}\leqslant C\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{3R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}d^{2}$

for either of the choices $w=(x,y)\cdot\nu$ and $w=R\nu_{y_{j}}$, and hence
(5) implies

(10) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/4}}|x|^{-2}d^{2}(x)\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{3R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}d^{2},\quad
C=C(\lambda,\Lambda,\varmathbb{C}).$

Since $\mathcal{M}(u_{j})=0$ and the $u_{j}$ are positive with small gradient,
and also
$d(\xi+u_{j}(\xi,\eta)\nu_{\varmathbb{C}_{0}}(\xi),\eta)=u_{j}(\xi,\eta)$, we
can use the Harnack inequality in balls of radius $R/20$, to give

$\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{3R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{4}}R\\}}d^{2}\leqslant
C\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/8}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{16}}R\\}}d^{2},$

and so (10) gives

(11) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/4}}|x|^{-2}d^{2}(x)\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/8}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{16}}R\\}}d^{2},\quad
C=C(\lambda,\Lambda,\varmathbb{C}).$

Then (1) follows from (11) after replacing $R$ by $4R$. (1) in particular
implies, for all sufficiently large $R$,

(12) $\left\\{\begin{aligned} &\int_{M\cap B_{R}\cap\\{(x,y):|x|<\delta
R\\}}d^{2}\,d\mu\leqslant C\delta^{2}\int_{M\cap
B_{R/2}}d^{2}\,d\mu\,\,\,\forall\delta\in(0,{\textstyle\frac{1}{2}}],\\\
&\int_{M\cap B_{R}}d^{2}\,d\mu\leqslant C\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<\frac{1}{4}R\\}}d^{2}\,d\mu,\end{aligned}\right.$

where $C=C(\lambda,\Lambda,\varmathbb{C})$.

So let $R_{k}\to\infty$ be arbitrary, $M_{k}=R_{k}^{-1}M$, $d_{k}=d|M_{k}$,
and

$u^{k}_{j}(x,y)=R_{k}^{-1}u_{j}(R_{k}x,R_{k}y)\big{/}\bigl{(}\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M_{k}\cap
B_{1}}d_{k}^{2}\bigr{)}^{1/2}.$

By virtue of (1), (6), and (12), we have a subsequence of $k$ (still denoted
$k$) such that $u^{k}_{j}$ converges locally in $\varmathbb{C}$ to a solution
$v_{j}\geqslant 0$ of $\mathcal{L}_{\varmathbb{C}}v_{j}=0$ with $v_{j}$
strictly positive for at least one $j$, and $\int_{\varmathbb{C}\cap
B_{R}}|x|^{-2}v_{j}^{2}\,d\mu<\infty$ for each $R>0$. Hence $v_{j}$ has a
representation of the form 3.20 on all of $\varmathbb{C}$, and then since
$v_{j}\geqslant 0$ we must have
$\smash{v_{j}=c_{j}r^{\gamma_{1}^{+}}\varphi_{1}}$ with $c_{j}\geqslant 0$ and
$c_{j}>0$ for at least one $j$. But then
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{\varmathbb{C}\cap
B_{R}}\sum_{j=1}^{q}v^{2}_{j}$ is constant, independent of $R$, and so (using
(12) again)

$\lim_{k\to\infty}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap
B_{2R_{k}}}d^{2}\bigg{/}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M_{k}\cap B_{R_{k}}}d^{2}=\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{2}}\sum_{j=1}^{q}v^{2}_{j}\bigg{/}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{\varmathbb{C}\cap
B_{1}}\sum_{j=1}^{q}v^{2}_{j}=1.$

Therefore, in view of the arbitrariness of the sequence $R_{k}$, for any
$\alpha\in(0,{\textstyle\frac{1}{2}}]$ we have

$2^{-\alpha}\leqslant\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}d^{2}\bigg{/}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}}d^{2}\leqslant 2^{\alpha}\,\,\,\forall\,R>R_{0},$

with $R_{0}=R_{0}(M,\lambda,\Lambda,\alpha)$, and 5 follows. $\Box$

##  6\. Proof of Theorem 1

According to Lemma 2 the unique tangent cone of $M$ at $\infty$ is
$\varmathbb{C}$ (with multiplicity $q$). Let $\alpha\in(0,1)$. By Lemma 5

6.1 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}d^{2}\,d\mu\leqslant R^{\alpha}$

for all sufficiently large $R$. Also by 4 either $\nu_{y}$ is identically zero
or there is a lower bound

6.2 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}|\nu_{y}|^{2}\geqslant R^{-\alpha}$

for all sufficiently large $R$, and by Lemma 4 with $w=\nu_{y_{j}}$ we have

6.3 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}|\nu_{y}|^{2}\leqslant C\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}|\nu_{y}|^{2},\quad
C=C(q,\lambda,\varmathbb{C}).$

By inequality (9) in the proof of Lemma 5 (with $R/2$ in place of $R$),

6.4 $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}|\nu_{y}|^{2}\leqslant
CR^{-2}\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2}.$

Combining 6.1, 6.2, 6.3 and 6.4, we then have

$R^{-\alpha}\leqslant\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B_{R}}|\nu_{y}|^{2}\leqslant CR^{-2}\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B_{R}}d^{2}\leqslant
R^{-2+\alpha}$

for all sufficiently large $R$ (depending on
$\alpha,q,\lambda,\varmathbb{C}$), which is impossible for $R>1$. Thus the
alternative that $\nu_{y}$ is identically zero on $M$ must hold, and so $M$ is
a cylinder $S\times\varmathbb{R}^{\ell}$.

##  7\. Proof of Corollary 1

We aim here to show that the hypotheses of Corollary 1 ensure that $M$ is
strictly stable as in 1.8, so that Corollary 1 is then implied by Theorem 1.

Before discussing the proof that $M$ is strictly stable, we need a couple of
preliminary results. First recall that, according to [HS85, Theorem 2.1],
since $\varmathbb{C}_{0}$ is minimizing there is a smooth embedded minimal
hypersurface $S\subset U_{+}$ ($U_{+}$ either one of the two connected
components of $\varmathbb{R}^{n+1}\setminus\overline{\varmathbb{C}}_{0}$),
with

7.1 ${\rm dist\,}(S,\\{0\\})=1$, ${\rm sing\,\,}S=\emptyset$ and $S$ is a
smooth radial graph; i.e. each ray $\bigl{\\{}tx:t>0\bigr{\\}}$ with $x\in
U_{+}$ meets $S$ transversely in just one point,

so in particular

7.2 $\displaystyle x\cdot\nu_{S}(x)>0\text{ for each $x\in S$, where $\nu_{S}$
is the smooth}$ unit normal of $S$ pointing away from $\varmathbb{C}_{0}$.

Furthermore, since $\varmathbb{C}_{0}$ is strictly stable and strictly
minimizing, [HS85, Theorem 3.2] ensures that, for
$R_{0}=R_{0}(\varmathbb{C}_{0})$ large enough, there is an $C^{2}$ function
$u$ defined on an open subset of $\Omega\subset\varmathbb{C}_{0}$ with
$\Omega\supset\bigl{\\{}x\in\varmathbb{C}_{0}:|x|>2R_{0}\bigr{\\}}$ and

7.3 $S\setminus
B^{n+1}_{R_{0}}=\bigl{\\{}x+u(x)\nu_{\varmathbb{C}_{0}}(x):x\in\Omega\bigr{\\}}\text{
with }u(x)=\kappa|x|^{\gamma_{1}}\varphi_{1}(|x|^{-1}x)+E(x),$

where $\kappa$ is a positive constant, $\nu_{\varmathbb{C}_{0}}$ is the unit
normal of $\varmathbb{C}_{0}$ pointing into $U_{+}$, $\varphi_{1}>0$ is as in
3.3 with $j=1$, and, for some $\alpha=\alpha(\varmathbb{C}_{0})>0$,

$\lim_{R\to\infty}\sup_{|x|>R}|x|^{k+|\gamma_{1}|+\alpha}|D^{k}E(x)|=0\text{
for }k=0,1,2.$

We claim that $S$ is strictly stable:

7.4 Lemma. If $S$ is as above, there is a constant
$\lambda=\lambda(\varmathbb{C}_{0})>0$ such that

$\lambda\int_{S}|x|^{-2}\zeta^{2}(x,y)\,d\mu(x,y)\leqslant\int_{S}\bigl{(}\bigl{|}\nabla_{S}\zeta\bigr{|}^{2}-|A_{S}|^{2}\zeta^{2}\bigr{)}\,d\mu,\quad\zeta\in
C_{c}^{1}(\varmathbb{R}^{n+1}),$

where $|A_{S}|$ is the length of the second fundamental form of $S$.

Proof: The normal part of the velocity vector of the homotheties $\lambda
S|_{\lambda>0}$ at $\lambda=1$ is $\psi=x\cdot\nu_{S}(x)\,(\,>0\text{
by\leavevmode\nobreak\ \ref{radial-graph}})$, and since the $\lambda S$ are
minimal hypersurfaces, this is a Jacobi function, i.e. a solution of

(1) $\Delta_{S}\psi+|A_{S}|^{2}\psi=0.$

By properties 7.2 and 7.3 we also have

(2) $C^{-1}|x|^{\gamma_{1}}\leqslant\psi(x)\leqslant C|x|^{\gamma_{1}}\text{
on $S$},\quad C=C(\varmathbb{C}_{0}).$

After a direct calculation and an integration by parts, (1) implies

(3)
$\int_{S}\psi^{2}|\nabla_{\\!S}\bigl{(}\zeta/\psi\bigr{)}|^{2}\,d\mu=\int_{S}\bigl{(}|\nabla_{\\!S}\zeta|^{2}-|A_{S}|^{2}\zeta^{2}\bigr{)}\,d\mu,$

and using the first variation formula 2.1 with $Z(x)=|x|^{-p-2}f^{2}(x)x$ and
noting that ${\rm div}_{S}x=n$, we have, after an application of the Cauchy-
Schwarz inequality,

(4) $\int_{S}|x|^{-p-2}f^{2}\leqslant
C\int_{S}|x|^{-p}|\nabla_{S}f|^{2},\,\,f\in
C^{1}_{c}(\varmathbb{R}^{n+1}),\quad p<n-2,\,\,C=C(p,n).$

7 is now proved by taking $f=\zeta/\psi$ and $p=2|\gamma_{1}|\,(<n-2)$, and
using (2), (3) and (4). $\Box$

As a corollary, any hypersurface sufficiently close to $S$ in the
appropriately scaled $C^{2}$ sense must also be strictly stable:

7.5 Corollary. For each $\theta\in(0,{\textstyle\frac{1}{2}}]$, there is
$\delta=\delta(\varmathbb{C}_{0},\theta)>0$ such that if $v\in C^{2}(S)$,
$M_{0}=\\{x+v(x)\nu_{S}(x):x\in S\\}\text{ and
}|x||\nabla^{2}_{S}v|+|\nabla_{S}v|+|x|^{-1}|v|\leqslant\delta\,\,\forall x\in
S$, then $M_{0}$ satisfies the inequality

$\lambda(1-\theta)\int_{M_{0}}|x|^{-2}\zeta^{2}\,d\mu\leqslant\int_{M_{0}}\bigl{(}\bigl{|}\nabla_{M_{0}}\zeta\bigr{|}^{2}-|A_{M_{0}}|^{2}\zeta^{2}\bigr{)}\,d\mu,\quad\zeta\in
C_{c}^{1}(\varmathbb{R}^{n+1}),$

where $|A_{M_{0}}|$ is the length of the second fundamental form of $M_{0}$,
and $\lambda$ is the constant of 7.

Proof: By 7, with $\tilde{\zeta}(x)=\zeta(x+v(x)\nu_{S}(x))$ for $x\in S$,

(1)
$\lambda\int_{S}|x|^{-2}\tilde{\zeta}^{2}\,d\mu\leqslant\int_{S}\bigl{(}\bigl{|}\nabla_{S}\tilde{\zeta}\bigr{|}^{2}-|A_{S}|^{2}\tilde{\zeta}^{2}\bigr{)}\,d\mu$

and for any $C^{1}$ function $f$ with compact support on $M_{0}$, with
$\tilde{f}(x)=f(x+v(x)\nu_{S}(x))$ for $x\in S$,

(2) $\left\\{\hskip 2.0pt\begin{aligned}
&{\textstyle\int}_{S}\tilde{f}\,d\mu={\textstyle\int}_{M_{0}}fJ\,d\mu\text{
with }|J-1|\leqslant C\delta\text{ (by the area formula)},\\\
&|\nabla_{S}\tilde{f}(x)-(\nabla_{M_{0}}f)(x+v(x)\nu_{S}(x)|\leqslant
C\delta|\nabla_{S}\tilde{f}(x)|\\\
&|A_{S}(x)-A_{M_{0}}(x+v(x)\nu_{S}(x))|<C|x|^{-1}\delta,\,\,\,C^{-1}|x|^{-1}\leqslant|A_{S}(x)|\leqslant
C|x|^{-1},\end{aligned}\right.$

where $C=C(S)$. By combining (1) and (2) we then have the required inequality
with $\theta=C\delta$. $\Box$

Next we need a uniqueness theorem for stationary integer multiplicity
varifolds with support contained in $\overline{U}_{+}$.

7.6 Lemma. If $M_{0}$ is a stationary integer multiplicity $n$-dimensional
varifold in $\varmathbb{R}^{n+1}$ with the properties that ${\rm
spt\,}M_{0}\subset\overline{U}_{+}$, ${\rm
spt\,}M_{0}\neq\overline{\varmathbb}{C}_{0}$, and
$\sup_{R>1}R^{-n}\mu(M_{0}\cap B_{R})<2\mu(\varmathbb{C}_{0}\cap B_{1})$, then
$M_{0}=\lambda S$ (with multiplicity $1$) for some $\lambda>0$, where $S$ is
as in 7.1.

Proof of 7: Let $C(M_{0})$ be a tangent cone of $M_{0}$ at $\infty$. Then by
the Allard compactness theorem $C(M_{0})$ is a stationary integer multiplicity
varifold with $\mu_{C(M_{0})}(B_{1})<2\mu(\varmathbb{C}_{0}\cap B_{1})$ and
${\rm spt\,}C(M_{0})\subset\overline{U}_{+}$. If ${\rm spt\,}C(M_{0})$
contains a ray of $\varmathbb{C}_{0}$ then, by the Solomon-White maximum
principle [SW89], either $C(M_{0})=\varmathbb{C}_{0}$ (with multiplicity one)
or else $C(M_{0})=\varmathbb{C}_{0}+V_{1}$, where $V_{1}$ is a non-zero
integer multiplicity cone with $\mu(V_{1}\cap
B_{1}(0))<\mu(\varmathbb{C}_{0}\cap B_{1})$ and support contained in
$\overline{U}_{+}$. On the other hand if ${\rm
spt\,}C(M_{0})\cap\varmathbb{C}_{0}=\emptyset$ then a rotation of ${\rm
spt\,}C(M_{0})$ has a ray in common with $\varmathbb{C}_{0}$ so by the same
argument applied to this rotation we still conclude that there is a stationary
cone $V_{1}$ with $\mu_{V_{1}}(B_{1})<\mu(\varmathbb{C}_{0}\cap B_{1})$ and
${\rm spt\,}V_{1}\subset\overline{U}_{+}$. But now by applying exactly the
same reasoning to $V_{1}$ we infer
$\mu_{V_{1}}(B_{1})\geqslant\mu(\varmathbb{C}_{0}\cap B_{1})$, a
contradiction. So $C(M_{0})=\varmathbb{C}_{0}$ and hence $\varmathbb{C}_{0}$,
with multiplicity one, is the unique tangent cone of $M_{0}$ at infinity.
Hence there is a $R_{0}>1$ and a $C^{2}(\varmathbb{C}_{0}\setminus B_{R_{0}})$
function $h$ with

(1) $\displaystyle\hskip 50.58878pt\text{$\sup_{x\in\varmathbb{C}_{0}\setminus
B_{R}}(|Dh(x)|+|x|^{-1}h(x))\to 0$ as $R\to\infty$},$ (2)
$\displaystyle\\{x+h(x)\nu_{\varmathbb{C}_{0}}(x):x\in\varmathbb{C}_{0}\setminus
B_{R_{0}}\\}\subset M_{0}$ $\displaystyle\hskip 72.26999pt\text{ and
}M_{0}\setminus\\{x+h(x)\nu_{\varmathbb{C}_{0}}(x):x\in\varmathbb{C}_{0}\setminus
B_{R_{0}}\\}\text{ is compact.}$

We also claim

(3) $0\notin M_{0}.$

Indeed otherwise the Ilmanen maximum principle [Ilm96] would give
$M_{0}\cap\varmathbb{C}_{0}\neq\emptyset$ and the above argument using the
Solomon-White maximum principle can be repeated, giving
$M_{0}=\varmathbb{C}_{0}$, contrary to the hypothesis that ${\rm
spt\,}M_{0}\neq\overline{\varmathbb}{C}_{0}$.

Observe next that if $r_{k}\to\infty$ the scaled minimal hypersurfaces
$r_{k}^{-1}M_{0}$ are represented by the graphs (taken off
$\varmathbb{C}_{0}$) of the functions $r_{k}^{-1}h(r_{k}x)\to 0$, and hence,
for any given $\omega_{0}\in\Sigma\,(=\varmathbb{C}_{0}\cap\partial B_{1})$,
the rescalings $(h(r_{k}\omega_{0}))^{-1}h(r_{k}r\omega)$ (which are bounded
above and below by positive constants on any given compact subset of
$\varmathbb{C}_{0}$ by (1) and the Harnack inequality) generate positive
solutions of the Jacobi equation $\mathcal{L}_{\varmathbb{C}_{0}}v=0$ on
$\varmathbb{C}_{0}$ as $k\to\infty$. But, by 3.1 and 3.2,
$(c_{1}r^{\gamma_{1}}+c_{2}r^{\gamma_{1}^{-}})\,\varphi_{1}(\omega)$ with
$c_{1},c_{2}\geqslant 0$ and $c_{1}+c_{2}>0$ are the only positive solutions
of $\mathcal{L}_{\varmathbb{C}_{0}}(\varphi)=0$ on all of $\varmathbb{C}_{0}$.
Thus in view of the arbitrariness of the sequence $r_{k}$, we have shown that

(4) $h(r\omega)=c(r)\varphi_{1}(\omega)+o(c(r))\text{ as $r\to\infty$,
uniformly for $\omega\in\Sigma$},$

and hence there are

$c_{-}(r)<c(r)<c_{+}(r)\text{ with
}c_{-}(r)\varphi_{1}(\omega)<h(r\omega)<c_{+}(r)\varphi_{1}(\omega)\text{ and
}c_{+}(r)/c_{-}(r)\to 1.$

Now, for suitable $R_{0}>0$, $S\setminus B_{R_{0}}$ ($S$ as in 7.1) also has a
representation of this form with some $\tilde{h}$ in place of $h$, where

$\tilde{h}(r\omega)=\kappa
r^{\gamma_{1}}\varphi_{1}(\omega)+o(r^{\gamma_{1}})\text{ as $r\to\infty$,
uniformly in $\omega$}$

and, similar to the choice of $c_{\pm}$, we can take $\tilde{c}_{\pm}(r)$ such
that

$\tilde{c}_{-}(r)\varphi_{1}(\omega)<\tilde{h}(r\omega)<\tilde{c}_{+}(r)\varphi_{1}(\omega)\text{
and }\tilde{c}_{+}(r)/\tilde{c}_{-}(r)\to 1\text{ as }r\to\infty.$

Now $\lambda S\setminus B_{\lambda R_{0}}$ can be represented by the
geometrically scaled function $\tilde{h}_{\lambda}$ with

(5)
$\tilde{h}_{\lambda}(r\omega)=\kappa\lambda^{1+|\gamma_{1}|}r^{-|\gamma_{1}|}\varphi_{1}+o(\kappa\lambda^{1+|\gamma_{1}|}r^{-|\gamma_{1}|}\varphi_{1})$

and we let $\lambda_{k}^{-}$ be the largest value of $\lambda$ such that
$\tilde{h}_{\lambda}(r_{k}\omega)\leqslant c_{-}(r_{k})\varphi_{1}(\omega)$
and $\lambda_{k}^{+}$ the smallest value of $\lambda$ such that
$\tilde{h}_{\lambda}(r_{k}\omega)\geqslant c_{+}(r_{k})\varphi_{1}(\omega)$.
Evidently there are then points $\omega_{\pm}\in\Sigma$ with
$\tilde{h}_{\lambda_{k}^{\pm}}(r_{k}\omega_{\pm})=c_{\pm}(r_{k})\varphi_{1}(\omega_{\pm})$
respectively. Also $M_{0}\cap\breve{B}_{r_{k}}$ must entirely lie in the
component $B_{r_{k}}\setminus\lambda_{k}^{+}S$ which contains
$\varmathbb{C}_{0}\cap B_{r_{k}}$; otherwise we could take the smallest
$\lambda>\lambda_{k}^{+}$ such that $M_{0}\cap B_{r_{k}}$ lies on in the
closure of that component of $B_{r_{k}}\setminus\lambda S$, and
$M_{0}\cap\breve{B}_{r_{k}}\cap\lambda S\neq\emptyset$, which contradicts the
maximum principle [SW89]. Similarly, $M_{0}\cap\breve{B}_{r_{k}}$ must
entirely lie in the component $B_{r_{k}}\setminus\lambda_{k}^{-}S$ which does
not contain $\varmathbb{C}_{0}\cap B_{r_{k}}$.

Thus $M_{0}\cap B_{r_{k}}$ lies between $\lambda_{k}^{+}S$ and
$\lambda_{k}^{-}S$ and by construction $\lambda_{k}^{+}/\lambda_{k}^{-}\to 1$,
and since $\lambda_{k}^{-}$ is bounded above, we then have a subsequence of
$k$ (still denoted $k$) such that $\lambda_{k}^{\pm}$ have a common (positive)
limit $\lambda$. So ${\rm spt\,}M_{0}\subset\lambda S$ and hence
$M_{0}=\lambda S$ with multiplicity $1$ by the constancy theorem and the fact
that $\sup_{R>1}R^{-n}\mu(M_{0}\cap B_{R})<2\mu(\varmathbb{C}_{0}\cap B_{1})$.
$\Box$

Finally we need to show that any $M$ satisfying the hypotheses of Corollary 1
with sufficiently small $\varepsilon_{0}$ must be strictly stable; then (as
mentioned at the beginning of this section) Corollary 1 follows from Theorem
1.

7.7 Lemma. For each $\alpha,\theta\in(0,1)$, there is
$\varepsilon_{0}=\varepsilon_{0}(\varmathbb{C},\alpha,\theta)\in(0,{\textstyle\frac{1}{2}}]$
such that if $M\subset U_{+}\times\varmathbb{R}^{\ell}$,
${\sup}_{R>1}R^{-n-\ell}\mu(M\cap
B_{R})\leqslant(2-\alpha)\mu(\varmathbb{C}\cap B_{1})$, and
${\sup}_{M}|\nu_{y}|<\varepsilon_{0}$, then $M$ is strictly stable in the
sense that

$(1-\theta)\lambda\int_{M}|x|^{-2}\zeta^{2}\,d\mu\leqslant\int_{M}\bigl{(}\bigl{|}\nabla_{M}\zeta\bigr{|}^{2}-|A_{M}|^{2}\zeta^{2}\bigr{)}\,d\mu,\quad\zeta\in
C_{c}^{1}(\varmathbb{R}^{n+1+\ell}),$

with $\lambda=\lambda(\varmathbb{C}_{0})>0$ as in 7.

Proof: For $y\in\varmathbb{R}^{\ell}$, define

(1) $M_{y}=\lambda_{y}^{-1}(M-(0,y)),\quad\lambda_{y}={\rm
dist\,}(M-(0,y),0).$

We claim that for each given $\delta>0$ the hypotheses of the lemma guarantee
that a strip of $M$ near a given slice $M_{z}=\\{(x,y)\in M:y=z\\}$ can be
scaled so that it is $C^{2}$ close to $S\times B^{\ell}_{1}$ ($S$ as in 7.2)
in the appropriately sense; more precisely, we claim that for each $\delta>0$
there is
$\varepsilon_{0}=\varepsilon_{0}(\varmathbb{C},\alpha,\theta,\delta)>0$ such
that the hypotheses of the lemma imply that for each
$z\in\varmathbb{R}^{\ell}$ there is $v_{z}\in C^{2}(\\{(x,y):x\in
S,\,|y|<1\\})$ such that

(2) $\left\\{\begin{aligned}
&M_{z}\cap\\{(x,y):|y|<1\\}=\bigl{\\{}(x+v_{z}(x,y)\nu_{S}(x),y):x\in
S,\,|y|<1\bigr{\\}}\\\
&\,|x||\nabla^{2}_{S\times\varmathbb{R}^{\ell}}v_{z}(x,y)|+|\nabla_{S\times\varmathbb{R}^{\ell}}v_{z}|(x,y)+|x|^{-1}|v_{z}(x,y)|\leqslant\delta\,\,\forall
x\in S,|y|<1.\end{aligned}\right.$

Otherwise this fails with $\varepsilon_{0}=1/k$ for each $k=1,2,\ldots$, so
there are minimal submanifolds $M_{k}$ such that the hypotheses hold with
$\varepsilon_{0}=1/k$ and with $M_{k}$ in place of $M$, yet there are points
$z_{k}\in M_{k}$ such that (2) fails with $z_{k},M_{k}$ in place of $z,M$.

We claim first that then there are fixed
$k_{0}=k_{0}(\delta)\in\\{1,2,\ldots\\},\,R_{0}=R_{0}(\delta)>1$ such that

(3) ${\rm
dist\,}((x,y),M_{z_{k}})<\delta|x|\,\,\forall(x,y)\in\varmathbb{C}\setminus
B_{R_{0}}^{n+1}\text{ with }|x|\geqslant|y|,\,\,\,k\geqslant k_{0}.$

Otherwise there would be a subsequence of $k$ (still denoted $k$) with

${\rm dist\,}((x_{k},y_{k}),M_{z_{k}})\geqslant\delta|x_{k}|$ with
$(x_{k},y_{k})\in\varmathbb{C}$, $|y_{k}|\leqslant|x_{k}|$ and
$|x_{k}|\to\infty$.

Then let

$\,\,\widetilde{\hskip-2.0ptM}_{k}=|x_{k}|^{-1}M_{z_{k}}.$

By the Allard compactness theorem, $\,\,\widetilde{\hskip-2.0ptM}_{k}$
converges in the varifold sense to a stationary integer multiplicity varifold
$V$ with support $M$ and density $\Theta$, where $M$ is a closed rectifiable
set, $\Theta$ is upper semi-continuous on $\varmathbb{R}^{n+1+\ell}$ and has
integer values $\mu$-a.e. on $M$, with $\Theta=0$ on
$\varmathbb{R}^{n+1+\ell}\setminus M$, and

(4) $M=\\{x:\Theta(x)\geqslant 1\\},$ (5) $\Theta(x)=\lim_{\rho\downarrow
0}\bigl{(}\omega_{n+\ell}\rho^{n+\ell}\bigr{)}^{-1}\int_{M\cap
B_{\rho}(x)}\Theta(\xi)\,d\mu(\xi)\,\,\,\forall x\in M,$

where $\omega_{n+\ell}$ is the volume of the unit ball in
$\varmathbb{R}^{n+\ell}$, $0\in
M\subset\overline{U}_{+}\times\varmathbb{R}^{\ell}$, and $M\cap
U_{+}\times\varmathbb{R}^{\ell}\neq 0$.

Taking $x_{n+1+j}=y_{j}$, so points in $\varmathbb{R}^{n+1+\ell}$ are written
$x=(x_{1},\ldots,x_{n+1+\ell})$, and letting
$\nu_{k}=(\nu_{k\,1},\ldots,\nu_{k\,n+1+\ell})$ be a unit normal for
$\,\,\widetilde{\hskip-2.0ptM}_{k}$ (so that the orthogonal projection of
$\varmathbb{R}^{n+1+\ell}$ onto $T_{x}\,\,\widetilde{\hskip-2.0ptM}_{k}$ has
matrix $(\delta_{ij}-\nu_{k\,i}\nu_{k\,j})$), the first variation formula for
$\,\,\widetilde{\hskip-2.0ptM}_{k}$ can be written

(6)
$\int_{\,\,\widetilde{\hskip-2.0ptM}_{k}}\sum_{i,j=1}^{n+1+\ell}(\delta_{ij}-\nu_{k\,i}\nu_{k\,j})D_{i}X_{j}\,d\mu=0,\,\,\,X_{j}\in
C^{1}_{c}(\varmathbb{R}^{n+1+\ell}),\,\,j=1,\ldots,n+1+\ell.$

Let $x_{0}\in M,\,\sigma>0$, and let $\tau\in
C^{\infty}(\varmathbb{R}^{n+1+\ell})$ with support $\tau\subset
B_{\sigma}(x_{0})$, and consider the function $T$ defined by

$T(x)=T(x_{1},\ldots,x_{n+1+\ell})=\int_{-\infty}^{x_{n+1+\ell}}\bigl{(}\tau(x_{1},\ldots,x_{n+\ell},t)-\tau(x_{1},\ldots,2\sigma+x_{n+\ell},t)\bigr{)}\,dt.$

Evidently $T\in C^{\infty}_{c}(\varmathbb{R}^{n+1+\ell})$ (indeed $T(x)=0$ on
$\varmathbb{R}^{n+1+\ell}\setminus K$, where $K$ is the cube
$\\{x\in\varmathbb{R}^{n+1+\ell}:|x_{i}-x_{0\,i}|<4\sigma\,\,\forall
i=1,\ldots,n+1+\ell\\}$). Therefore we can use (6) with
$X(x)=T(x)e_{n+1+\ell}$, giving

$\int_{\,\,\widetilde{\hskip-2.0ptM}_{k}}\bigl{(}1-\nu_{k\,n+1+\ell}^{2}\bigr{)}\bigl{(}\tau(x)-\tau(x+2\sigma
e_{n+1+\ell})\bigr{)}\,d\mu=\int_{\,\,\widetilde{\hskip-2.0ptM}_{k}}\sum_{i=1}^{n+\ell}\nu_{k\,n+1+\ell}\nu_{k\,i}D_{i}T\,d\mu,$

and hence

$\Bigl{|}\int_{\,\,\widetilde{\hskip-2.0ptM}_{k}}\tau(x)\,d\mu-\int_{\,\,\widetilde{\hskip-2.0ptM}_{k}}\tau(x+2\sigma
e_{n+1+\ell})\,d\mu\Bigr{|}\leqslant C/k.$

Using the fact that varifold convergence of
$\,\,\widetilde{\hskip-2.0ptM}_{k}$ implies convergence of the corresponding
mass measures $\mu\hbox{ {\vrule height=6.25963pt}{\leaders\hrule\hskip
5.69046pt} }\,\,\widetilde{\hskip-2.0ptM}_{k}$ to $\mu\hbox{ {\vrule
height=6.25963pt}{\leaders\hrule\hskip 5.69046pt} }\Theta$, we then have

(7) $\int_{M}\tau(x)\,\Theta(x)d\mu(x)=\int_{M}\tau(x+2\sigma
e_{n+1+\ell})\,\Theta(x)d\mu(x).$

Taking $\rho\in(0,\sigma)$ and replacing $\tau$ by a sequence $\tau_{k}$ with
$\tau_{k}\downarrow\chi_{B_{\rho}(x_{0})}$ (the indicator function of the
closed ball $\smash{B_{\rho}(x_{0})}$), we then conclude

$\int_{M\cap B_{\rho}(x_{0})}\,\Theta\,d\mu=\int_{M\cap B_{\rho}(x_{0}+2\sigma
e_{n+1+\ell})}\ \Theta\,d\mu$

and hence by (5)

$\Theta(x_{0})=\Theta(x_{0}+2\sigma e_{n+1+\ell}).$

In view of the arbitrariness of $\sigma$ this shows that $\Theta(x)$ is
independent of the variable $x_{n+1+\ell}$, and the same argument shows that
$\Theta(x)$ is also independent of $x_{n+1+j}$, $j=1,\ldots,\ell-1$. Thus, by
(4), $M$ is cylindrical: $M=M_{0}\times\varmathbb{R}^{\ell}$, where
$R^{-n}\int_{M_{0}\cap B_{R}}\Theta\,d\mu<2\mu(\varmathbb{C}_{0}\cap B_{1})$,
$0\in M_{0}\subset\overline{U}_{+}$, and $M_{0}\cap U_{+}\neq\emptyset$. Hence
$M_{0}=\lambda S$ for some $\lambda>0$ by virtue of Lemma 7, contradicting the
fact that $0\in M_{0}$. So (3) is proved, and (3) together with the Allard
regularity theorem implies that
$M_{z_{k}}\cap\bigl{(}\bigl{\\{}(x,y)\in\varmathbb{R}^{n+1+\ell}\setminus
B_{2R_{0}}:|y|<{\textstyle\frac{1}{2}}|x|\bigr{\\}}\bigr{)}$ is $C^{2}$ close
to $\varmathbb{C}$, in the sense that there is are $C^{2}$ functions $v_{k}$
on a domain in $\varmathbb{C}$ with

(8) $\displaystyle{\rm
graph\,}v_{k}=M_{z_{k}}\cap\bigl{(}\bigl{\\{}(x,y)\in\varmathbb{R}^{n+1+\ell}\setminus
B_{2R_{0}}:|y|<{\textstyle\frac{1}{2}}|x|\bigr{\\}}\bigr{)}$
$\displaystyle\hskip 122.85876pt\text{ and
}|x|^{-1}|v_{k}|+|\nabla_{\varmathbb{C}}v_{k}|+|x||\nabla^{2}_{\varmathbb{C}}v_{k}|<C\delta.$

Next, exactly the same compactness discussion can be applied with $M_{z_{k}}$
in place of $\,\,\widetilde{\hskip-2.0ptM}_{k}$, giving a cylindrical varifold
limit $M=M_{0}\times\varmathbb{R}^{\ell}$, $M_{0}$ a closed subset of
$\overline{U}_{+}$ with density $\Theta\geqslant 1$, but this time ${\rm
dist\,}(M_{0},0)=1$. Hence $\Theta\equiv 1$ and $M_{0}=S$ by 7. But then the
Allard regularity theorem guarantees that the convergence of $M_{z_{k}}$ to
$M$ is smooth and hence, by virtue of (8), (2) holds with $M_{z_{k}}$ in place
of $M$, a contradiction. Thus (2) is proved.

Then by Corollary 7, for small enough $\delta=\delta(\varmathbb{C},\theta)$,
for each $y\in\varmathbb{R}^{\ell}$

$(1-\theta/2)\lambda\int_{M_{y}}|x|^{-2}\zeta_{y}^{2}(x)\,d\mu(x)\leqslant\int_{M_{y}}\bigl{(}\bigl{|}\nabla_{M_{y}}\zeta_{y}(x)\bigr{|}^{2}-|A_{M_{y}}|^{2}\zeta_{y}^{2}(x)\bigr{)}\,d\mu(x),$

$\zeta\in C_{c}^{1}(\varmathbb{R}^{n+1})$, where
$M_{y}=\\{x\in\varmathbb{R}^{n+1}:(x,y)\in M\\}$, $\zeta_{y}(x)=\zeta(x,y)$,
and $|A_{M_{y}}|$ is the length of the second fundamental form of $M_{y}$.

Since

$||A_{M_{y}}|^{2}(x)-|A_{M}|^{2}(x,y)|\leqslant
C\delta/|x|^{2},\,\,\,|A_{M_{y}}|^{2}(x)\leqslant C/|x|^{2}$

by (2), the proof is completed by taking $\delta$ small enough, integrating
with respect to $y\in\varmathbb{R}^{\ell}$, and using the coarea formula
together with (2). $\Box$

##  8\. A Boundary Version of Theorem 1 and Corollary 1

Since it will be needed in [Sim21b], we here also want to discuss a version of
Theorem 1 which is valid in case $M$ has a boundary.

8.1 Theorem. Suppose $M\subset\varmathbb{R}^{n+1}\times\\{y:y_{\ell}\geqslant
0\\}$ is a complete embedded minimal hypersurface-with-boundary, with

$None$ $\left\\{\hskip 2.0pt\begin{aligned} &\partial
M=S\times\\{y:y_{\ell}=0\\},\\\ \vskip-1.0pt\cr&|\nu_{y_{\ell}}|<1\text{ on
}S,\,\,\\\ \vskip-1.0pt\cr&{\sup}_{R>1}R^{-n-\ell}\mu(M\cap
B_{R})<2\mu(\varmathbb{C}\cap\bigl{\\{}(x,y)\in
B_{1}:y_{\ell}>0\bigr{\\}}),\text{ and }\\\ \vskip-1.0pt\cr&M\subset
U_{\lambda}\times\\{y:y_{\ell}\geqslant 0\\}\end{aligned}\right.$

for some $\lambda\geqslant 1$, where $U_{\lambda}$ denotes the component of
$U_{+}\setminus\lambda S$ with $\partial
U_{\lambda}=\overline{\varmathbb}{C}_{0}\cup\lambda S$.

Suppose also that $M$ is strictly stable in the sense there is $\lambda>0$
such that that the inequality  _1.8_ holds for all $\zeta\in
C^{1}_{c}(M\cap\breve{B}_{R})$ with $e_{n+1+\ell}\cdot\nabla_{M}\zeta=0$ on
$\partial M$ and for all $R>0$.

Then

$M=S\times\\{y:y_{\ell}\geqslant 0\\}.$

Proof: Since $M\subset U_{\lambda}$, the Allard compactness theorem, applied
to the rescalings $\tau M,\,\tau\downarrow 0$, plus the constancy theorem,
tells us that $M$ has a tangent cone at $\infty$ which is
$\varmathbb{C}_{0}\times\\{y:y_{\ell}\geqslant 0\\}$ with some integer
multiplicity $k\geqslant 1$, and then the condition
$\sup_{R>1}R^{-n-1}\mu(M\cap B_{R})<2\mu(\varmathbb{C}\cap\bigl{\\{}(x,y)\in
B_{1}:y>0\bigr{\\}})$ implies $k=1$. Thus $M$ has
$\varmathbb{C}_{0}\times\\{y:y_{\ell}\geqslant 0\\}$ with multiplicity 1 as
its unique tangent cone at $\infty$.

We claim that $\nu_{y_{\ell}}$ satisfies the free boundary condition

(1) $e_{n+1+\ell}\cdot\nabla_{M}\nu_{y_{\ell}}=0\text{ at each point of
$\partial M$. }$

Indeed if $\Sigma\subset\varmathbb{R}^{N}\times[0,\infty)$ is any minimal
hypersurface-with-boundary with smooth unit normal
$\nu=(\nu_{1},\ldots,\nu_{N+1})$ and $|\nu_{N+1}|\neq 1$ on $\partial\Sigma$
(i.e. $\Sigma$ intersects the hyperplane $x_{N+1}=0$ transversely), and if
$\partial\Sigma$ is a minimal hypersurface in
$\varmathbb{R}^{N}\times\\{0\\}$, then
$e_{N+1}\cdot\nabla_{\Sigma}\nu_{N+1}=0$ on $\partial\Sigma$, as one readily
checks by using the fact that the mean curvature (i.e. trace of second
fundamental form) of $\Sigma$ and $\partial\Sigma$ are both zero at each point
of $\partial\Sigma$.

We claim $\nu_{y_{\ell}}(x,y)=0\,\,\forall(x,y)\in M$. To check this, first
observe that there is a version of 4 which is valid in the half-space
$y_{\ell}\geqslant 0$ in the case when $w$ has free boundary condition
$e_{n+1+\ell}\cdot\nabla_{M}w=0$ on $\partial M$; indeed the proof of 4 goes
through with little change—the linear solutions $v$ of
$\mathcal{L}_{\varmathbb{C}}v=0$ obtained in the proof being defined on the
half-space $\varmathbb{C}_{0}\times\\{y:y_{\ell}\geqslant 0\\}$ and having the
free boundary condition $e_{n+1+\ell}\cdot\nabla_{\varmathbb{C}}v=0$. So $v$
extends to a solution of $\mathcal{L}_{\varmathbb{C}}v=0$ on all of
$\varmathbb{C}$ by even reflection and the rest of the argument is unchanged.

In particular, since $\nu_{y_{\ell}}$ has free boundary condition $0$ by (1),
we have an analogue of Lemma 4 in the half-space, giving

(2) $R^{-\alpha}\leqslant C\hskip
1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap B^{+}_{R}}\nu_{y_{\ell}}^{2}$

for each $\alpha\in(0,1)$, where
$B^{+}_{R}=B_{R}\cap\\{(x,y):y_{\ell}\geqslant 0\\}$, and, since
$\nu_{y_{\ell}}\leqslant 1$ there is a bound
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{M\cap
B^{+}_{R}}\nu_{y_{\ell}}^{2}\leqslant CR^{n-2-\beta}$, so by Corollary 4

(3) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B^{+}_{R}}\nu_{y_{\ell}}^{2}\leqslant
CR^{-\ell-2-\beta_{1}}\int_{B^{+}_{R/2}\setminus\\{(x,y):|x|<{\scriptsize\frac{1}{3}}R\\}}\nu_{y_{\ell}}^{2},$

where $C=C(\varmathbb{C}_{0},\alpha)$. Since $M\subset
U_{\lambda}\times[0,\infty)$ also we have $d(x)\leqslant
C\lambda\min\\{1,\,|x|^{\gamma_{1}}\\}$ for all $x\in\varmathbb{R}^{n}$ and
hence

(4) $\hskip 1.0pt\text{\small$-$}\hskip-10.6pt\int_{M\cap
B^{+}_{R}}d^{2}\leqslant C\lambda^{2},\quad C=C(\varmathbb{C}_{0}).$

(Notice that here we do not need growth estimates for
$\text{\scriptsize$-$}\hskip-6.85pt{\textstyle\int}_{B_{R}}d^{2}$ as in §5
because we are now assuming $d\leqslant
C\lambda\min\\{1,\,|x|^{\gamma_{1}}\\}$.) Then the proof that $\nu_{y_{\ell}}$
must be identically zero is completed using (2), (3), and (4) analogously to
the proof of non-boundary version of Theorem 1.

So $\nu_{y_{\ell}}$ is identically zero on $M$. This completes the proof in
the case $\ell=1$ and shows that, for $\ell\geqslant 2$, by even reflection
$M$ extends to a minimal submanifold $\,\,\widetilde{\hskip-2.0ptM}$ (without
boundary)

$\,\,\widetilde{\hskip-2.0ptM}=M\cup\bigl{\\{}(x,y_{1},\ldots,y_{\ell-1},-y_{\ell}):(x,y)\in
M\bigr{\\}}\subset U_{+}\times\varmathbb{R}^{\ell},$

which is translation invariant by translations in the direction of
$e_{n+1+\ell}$. Then $\,\,\widetilde{\hskip-2.0ptM}$ is strictly stable and
Theorem 1 applies, giving $M=S\times\varmathbb{R}^{\ell-1}\times[0,\infty)$,
as claimed. $\Box$

8.2 Corollary. There is $\delta=\delta(\varmathbb{C},\lambda)>0$ such that if
$M$ satisfies 8 $({\ddagger})$ and $\sup|\nu_{y}|<\delta$, then $M$
automatically satisfies the strict stability hypothesis in 8, and hence
$M=S\times\\{y:y_{\ell}\geqslant 0\\}$.

Proof: With $\,\,\widetilde{\hskip-2.0ptM}_{k}$, $M_{z_{k}}$ as in the proof
of 7, with only minor modifications to the present situation when $\partial
M=S\times\\{0\\}$, we have $\,\,\widetilde{\hskip-2.0ptM}_{k}$ and $M_{z_{k}}$
both have cylindrical limits $S\times\varmathbb{R}^{\ell}$ or
$S\times\\{y\in\varmathbb{R}^{\ell}:y_{\ell}\geqslant K\\}$ for suitable $K$,
and so (using the Allard boundary regularity theorem in the latter case) the
$y=$ const. slices of $M$ have the $C^{2}$ approximation property 7(2). Hence,
by integration over the slices as in 7, $M$ is strictly stable in the sense
that 1.8 holds for all $\zeta\in C^{1}_{c}(\breve{B}_{R}^{+})$, where
$\breve{B}_{R}^{+}=\\{(x,y):|(x,y)|<R\text{ and }y_{\ell}\geqslant 0\\}$.
$\Box$

## Appendix: Analyticity of $\beta$-harmonic functions

For $\rho>0$, $r_{0}\geqslant 0$ and $y_{0}\in\varmathbb{R}^{\ell}$, let

$\displaystyle
B_{\rho}^{+}(r_{0},y_{0})=\\{(r,y)\in\varmathbb{R}\times\varmathbb{R}^{\ell}:r\geqslant
0,\,\,(r-r_{0})^{2}+|y-y_{0}|^{2}\leqslant\rho^{2}\\},$
$\displaystyle\breve{B}^{+}_{\rho}(r_{0},y_{0})=\\{(r,y)\in\varmathbb{R}\times\varmathbb{R}^{\ell}:r>0,\,\,(r-r_{0})^{2}+|y-y_{0}|^{2}<\rho^{2}\\},$

and $B^{+}_{\rho},\,\breve{B}^{+}_{\rho}$ will be used as abbreviations for
$B^{+}_{\rho}(0,0),\,\breve{B}^{+}_{\rho}(0,0)$ respectively.

Our aim is to prove real analyticity extension across $r=0$ of
$\beta$-harmonic functions, i.e. solutions $u\in
C^{\infty}(\breve{B}_{1}^{+})$ of

A.1 $r^{-\gamma}\frac{\partial}{\partial r}\bigl{(}r^{\gamma}\frac{\partial
u}{\partial r}\bigr{)}+\Delta_{y}u=0,$

where $\gamma=1+\beta\,\,(\,>1)$ and
$\Delta_{y}u={\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}D_{y_{j}}u$, assuming

A.2
$\int_{B^{+}_{\rho}}(u_{r}^{2}+|u_{y}|^{2})\,\,r^{\gamma\\!}drdy<\infty\,\,\,\forall\,\rho<1.$

In fact we show under these conditions that $u$ can be written as a convergent
series of homogeneous $\beta$-harmonic polynomials in $\\{(r,y):r\geqslant
0,\,r^{2}+|y|^{2}<1\\}$ with the convergence uniform in $B_{\rho}^{+}$ for
each $\rho<1$.

Of course all solutions of A.1 satisfying A.2 are automatically real-analytic
in $\breve{B}^{+}_{1}$ because the equation is uniformly elliptic with real-
analytic coefficients in each closed ball $\subset\breve{B}^{+}_{1}$. Also, if
$\gamma$ is an integer $\geqslant 1$ then the operator in A.1 is just the
Laplacian in $\varmathbb{R}^{1+\gamma+\ell}$, at least as it applies to
functions $u=u(x,y)$ on $\varmathbb{R}^{1+\gamma+\ell}$ which can be expressed
as a function of $r=|x|$ and $y$, so in this case smoothness across $r=0$ is
Weyl’s lemma and analyticity is then standard.

To handle the case of general $\gamma>1$, first note that by virtue of the
calculus inequality $\int_{0}^{a}r^{\gamma-2}f^{2}\,dr\leqslant
C\int_{0}^{a}r^{\gamma}(f^{\prime}(r))^{2}\,dr$ for any $f\in C^{1}(0,a]$ with
$f(a)=0$ (which is proved simply by using the identity
$\int_{0}^{a}\bigl{(}r^{\gamma-1}(\min\\{|f|,k\\})^{2}\bigr{)}^{\prime}\,dr=0$
and then using Cauchy-Schwarz inequality and letting $k\to\infty$), we have

A.3 $\int_{B^{+}_{\rho}}r^{-2}\zeta^{2}\,d\mu_{+}\leqslant
C\int_{B^{+}_{\rho}}\bigl{(}D_{r}\zeta\bigr{)}^{2}\,d\mu_{+},\quad
C=C(\gamma)$

for any $\rho\in[{\textstyle\frac{1}{2}},1)$ and any Lipschitz $\zeta$ on
$B^{+}_{1}$ with support contained in $B^{+}_{\rho}$.

Next observe that $u$ satisfies the weak form of A.1:

A.4 $\int_{\breve{B}_{1}^{+}}(u_{r}\zeta_{r}+u_{y}\cdot\zeta_{y})\,d\mu_{+}=0$

for any Lipschitz $\zeta$ with support in $B^{+}_{\rho}$ for some $\rho<1$,
where, here and subsequently,

$d\mu_{+}=r^{\gamma}drdy$

and subscripts denote partial derivatives:

$u_{r}=D_{r}u,\,\,\,u_{y}=D_{\\!y}u=(D_{\\!y_{1}}u,\ldots,D_{\\!y_{\ell}}u).$

A.4 is checked by first observing that it holds with
$\varphi_{\sigma}(r)\zeta(r,y)$ in place of $\zeta(r,y)$, where
$\varphi_{\sigma}(r)=0$ for $r<\sigma/2$, $\varphi_{\sigma}(r)=1$ for
$r>\sigma$, $|\varphi_{\sigma}^{\prime}(r)|<C/\sigma$, and then letting
$\sigma\downarrow 0$ and using A.2.

Similarly since $r^{-\gamma}\frac{\partial}{\partial
r}(r^{\gamma}\frac{\partial u^{2}}{\partial
r})+{\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}D_{y_{j}}u^{2}=2(u_{r}^{2}+|u_{y}|^{2})\geqslant
0$, we can check using A.2 and A.3 that $u^{2}$ is a weak subsolution, meaning
that

A.5
$\int_{\breve{B}_{1}^{+}}\bigl{(}(u^{2})_{r}\zeta_{r}+(u^{2})_{y}\cdot\zeta_{y}\bigr{)}\,d\mu_{+}\leqslant
0,$

for any non-negative Lipschitz function $\zeta$ on $B^{+}_{1}$ with support
$\subset B^{+}_{\rho}$ for some $\rho<1$.

Next we note that if $\rho\in[{\textstyle\frac{1}{2}},1)$, $r_{0}\geqslant 0$
and $B_{\rho}^{+}(r_{0},y_{0})\subset B^{+}_{1}$, then

A.6 $|u(r_{0},y_{0})|\leqslant
C\bigl{(}\rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho}(r_{0},y_{0})}u^{2}\,d\mu_{+}\bigr{)}^{1/2},\quad
C=C(\gamma,\ell).$

To check this, first observe that if
$B^{+}_{\sigma}(r_{0},y_{0})\subset\breve{B}_{1}^{+}$ and $r_{0}>\sigma$ then
the equation A.1 is uniformly elliptic divergence form with smooth
coefficients on $B_{\sigma/2}(r_{0},y_{0})$, so we can use standard elliptic
estimates for $u$ to give

A.7 $|u(r_{0},y_{0})|^{2}\leqslant
C\sigma^{-\gamma-1-\ell}\int_{B^{+}_{\sigma}(r_{0},y_{0})}u^{2}\,d\mu_{+}$

with $C=C(\gamma,\ell)$. So now assume $B_{\rho}^{+}(r_{0},y_{0})\subset
B^{+}_{1}$. If $r_{0}>\rho/4$ we can use A.7 with $\sigma=\rho/4<r_{0}$ to
give A.6, while on the other hand if $r_{0}\leqslant\rho/4$ then we can first
take $\sigma=r_{0}/2$ in A.7 to give

A.8 $|u(r_{0},y_{0})|^{2}\leqslant
Cr_{0}^{-\ell-1-\gamma}\int_{B^{+}_{r_{0}/2}(r_{0},y_{0})}u^{2}\,d\mu_{+}\leqslant
Cr_{0}^{-\ell-1-\gamma}\int_{B^{+}_{2r_{0}}(0,y_{0})}u^{2}\,d\mu_{+},$

and then observe, using a straightforward modification of the relevant
argument for classical subharmonic functions to the present case of the
$\beta$-subharmonic function $u^{2}$ as in A.5,

$\text{$\sigma^{-\ell-1-\gamma}\int_{B^{+}_{\sigma}(0,y_{0})}u^{2}\,d\mu_{+}$
is an increasing function of $\sigma$ for $\sigma\in(0,\rho/2]$}.$

So from A.8 we conclude

$|u(r_{0},y_{0})|^{2}\leqslant
C\rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho/2}(0,y_{0})}u^{2}\,d\mu_{+}\leqslant
C\rho^{-\ell-1-\gamma}\int_{B^{+}_{\rho}(r_{0},y_{0})}u^{2}\,d\mu_{+}$

where $C=C(\gamma,\ell)$. Thus A.6 is proved.

For $\rho\in[{\textstyle\frac{1}{2}},1)$,
$kh\in\bigl{(}-(1-\rho),1-\rho\bigr{)}\setminus\\{0\\}$, and
$k\in\\{1,2,\ldots\\}$, let $u_{h}^{(k)}$ (defined on $\breve{B}^{+}_{\rho}$)
denote the vector of $k$-th order difference quotients with respect to the
$y$-variables; so for example

$\displaystyle u_{h}^{(1)}$
$\displaystyle=\bigl{(}h^{-1}(u(x,y+he_{1})-u(x,y)),\ldots,h^{-1}(u(x,y+he_{\ell})-u(x,y))\bigr{)},$
$\displaystyle u_{h}^{(2)}$
$\displaystyle=\bigl{(}h^{-1}(u_{h}^{(1)}(x,y+he_{1})-u_{h}^{(1)}(x,y)),\ldots,h^{-1}(u_{h}^{(1)}(x,y+he_{\ell})-u_{h}^{(1)}(x,y))\bigr{)},$

and generally, for $k|h|<1-\rho$,

$\displaystyle
u_{h}^{(k)}(x,y)=h^{-1}\bigl{(}(u_{h}^{(k-1)}(x,y+he_{1})-u_{h}^{(k-1)}(x,y)),\ldots,$
$\displaystyle\hskip
166.2212pt(u_{h}^{(k-1)}(x,y+he_{\ell})-u_{h}^{(k-1)}(x,y))\bigr{)},$

(which is a function with values in $\varmathbb{R}^{\ell^{k}}$). For
notational convenience we also take

$u^{(0)}_{h}(x,y)=u(x,y).$

Then, replacing $\zeta$ in A.4 by $\zeta_{-h}^{(k)}$ and changing variables
appropriately (i.e. “integration by parts” for finite differences instead of
derivatives),

$\int_{\breve{B}_{1}^{+}}\bigl{(}D_{r}u^{(k)}_{h}D_{r}\zeta+{\textstyle\sum}_{j=1}^{\ell}D_{y_{j}}u^{(k)}_{h}\,D_{y_{j}}\zeta\bigr{)}\,d\mu_{+}=0,$

for all Lipschitz $\zeta$ on $B_{1}^{+}$ with support $\subset B^{+}_{\rho}$
and for $k|h|<1-\rho$. Replacing $\zeta$ by $\zeta^{2}u_{h}^{(k)}$ gives

$\displaystyle\int_{\breve{B}_{1}^{+}}\bigl{(}|D_{r}u_{h}^{(k)}|^{2}+|D_{y}u_{h}^{(k)}|^{2}\bigr{)}\zeta^{2}$
$\displaystyle\hskip
72.26999pt=-2\smash[t]{\int_{\breve{B}_{1}^{+}}}\bigl{(}\zeta u_{h}^{(k)}\cdot
D_{r}u_{h}^{(k)}D_{r}\zeta+\zeta u_{h}^{(k)}\cdot
D_{r}u_{h}^{(k)}D_{y_{j}}\zeta D_{y_{j}}\zeta\bigr{)}\,d\mu_{+},$

so by Cauchy-Schwarz

$\int_{\breve{B}_{1}^{+}}\bigl{(}|D_{r}u_{h}^{(k)}|^{2}+|D_{y}u_{h}^{(k)}|^{2}\bigr{)}\zeta^{2}\,d\mu_{+}\leqslant
4\int_{\breve{B}_{1}^{+}}|u_{h}^{(k)}|^{2}|D\zeta|^{2}\,d\mu_{+},$

and by A.3 we then have

A.9
$\int_{\breve{B}_{1}^{+}}\bigl{(}r^{-2}|u_{h}^{(k)}|^{2}+|D_{r}u_{h}^{(k)}|^{2}+|D_{y}u_{h}^{(k)}|^{2}\bigr{)}\zeta^{2}\,d\mu_{+}\leqslant
C\int_{\breve{B}_{1}^{+}}|u_{h}^{(k)}|^{2}|D\zeta|^{2}\,d\mu_{+},$

where $C=C(\gamma,\ell)$. Now let $u^{(k)}=\lim_{h\to 0}u_{h}^{(k)}$ (i.e.
$u^{(k)}$ is the array of all mixed partial derivatives of order $k$ with
respect to the $y$ variables defined inductively by $u^{(0)}=u$ and
$u^{(k+1)}=D_{y}u^{(k)}$). Then A.9 gives

A.10
$\int_{\breve{B}_{1}^{+}}\bigl{(}r^{-2}|u^{(k)}|^{2}+|D_{r}u^{(k)}|^{2}+|u^{(k+1)}|^{2}\bigr{)}\zeta^{2}\leqslant
C\int_{\breve{B}_{1}^{+}}|u^{(k)}|^{2}|D\zeta|^{2}\,d\mu_{+}$

for each $\zeta\in C^{1}(B^{+}_{\rho})$ with support in $B^{+}_{\theta\rho}$
for some $\theta\in[{\textstyle\frac{1}{2}},1)$ and for each $k$ such that the
right side is finite. By A.2 the right side is finite for $k=0,1$, and taking
$k=1$ in A.10 then implies the right side is also finite with $k=2$.
Proceeding inductively we see that in fact that the right side is finite for
each $k=0,1,\ldots$, so A.10 is valid and all integrals are finite for all
$k=1,2,\ldots$.

Let $k\in\\{1,2,\ldots\\}$ and $(r_{0},y_{0})\in B^{+}_{1}$ with
$r_{0}\geqslant 0$. Then if $B^{+}_{\rho}(r_{0},y_{0})\subset B^{+}_{1}$ and
we let

$\rho_{j}=\rho-{\textstyle\frac{j}{k}}\rho/2,\,\,j=0,\ldots,k-1,$

and, applying A.10 with $k=j$ and

$\zeta=1\text{ on }B^{+}_{\rho_{j+1}}(r_{0},y_{0}),\,\,\zeta=0\text{ on
}\varmathbb{R}^{1+\ell}\setminus B^{+}_{\rho_{j}}(r_{0},y_{0}),\text{ and
}|D\zeta|\leqslant 3k,$

we obtain

$\int_{\breve{B}^{+}_{\rho_{j+1}}(r_{0},y_{0})}|u^{(j+1)}|^{2}\,d\mu_{+}\leqslant
C\rho^{-2}k^{2}\int_{\breve{B}^{+}_{\rho_{j}}(r_{0},y_{0})}|u^{(j)}|^{2}\,d\mu_{+}$

with $C=C(\gamma,\ell)$. By iteration this gives

$\int_{\breve{B}_{\rho/2}^{+}(r_{0},y_{0})}|u^{(k)}|^{2}\,d\mu_{+}\leqslant
C^{k}\rho^{-2k}(k!)^{2}\int_{\breve{B}_{\rho}^{+}(r_{0},y_{0})}|u|^{2}\,d\mu_{+}$

with suitable $C=C(\gamma,\ell)$ (independent of $k$), and then by A.6 with
$u^{(k)}$ in place of $u$ and $\rho/2$ in place of $\rho$ we obtain

$|u^{(k)}(r_{0},y_{0})|^{2}\leqslant
C^{k}\rho^{-2k}(k!)^{2}\rho^{-\gamma-1-\ell}\int_{\breve{B}^{+}_{\rho}(r_{0},y_{0})}|u|^{2}\,d\mu_{+},$

with $C=C(\gamma,\ell)$ (independent of $k$ and $\rho$). In view of the
arbitrariness of $\rho,r_{0},y_{0}$ this implies

A.11 $\sup_{B^{+}_{\rho/2}}|u^{(k)}|^{2}\leqslant
C^{k}(k!)^{2}\int_{\breve{B}^{+}_{\rho}}|u|^{2}\,d\mu_{+},\text{ for each
$\rho\in(0,1)$},$

where $C=C(\gamma,\ell,\rho)$.

Next let $L_{r}$ be the second order operator defined by

$L_{r}(f)=r^{-\gamma}\frac{\partial}{\partial
r}\bigl{(}r^{\gamma}\frac{\partial f}{\partial r}\bigr{)},$

so that A.1 says $L_{r}u=-\Delta_{y}u$, where
$\Delta_{y}u={\textstyle\sum}_{j=1}^{\ell}D^{2}_{y_{j}}u$, and by repeatedly
differentiating this identity with respect to the $y$ variables, we have

$L_{r}u^{(k)}(r,y)=-\Delta_{y}u^{(k)},$

and hence

A.12 $L_{r}^{j}u^{(k)}=-(\Delta_{y})^{j}u^{(k)}.$

for each $j,k=0,1,\ldots$. In particular, since $|\Delta_{y}^{j}f|\leqslant
C^{j}|f^{(2j)}|$ with $C=C(\ell)$,

A.13 $|L_{r}^{j}u^{(k)}|^{2}\leqslant C^{2j}|u^{(k+2j)}|^{2}$

for each $j,k=0,1,2,\ldots$, where $C=C(\ell)$

Next we note that, since $|u^{(k)}|$ is bounded on $B^{+}_{\rho}$ for each
$\rho<{\textstyle\frac{1}{2}}$ by A.11, for small enough $r>0$ and
$|y|<{\textstyle\frac{1}{2}}$ we can apply elliptic estimates to give
$|D_{r}u^{(k)}|<C/r$ with $C$ fixed independent of $r$ (but depending on $k$),
and since $\gamma>1$ we have $|r^{\gamma}D_{r}u^{(k)}(r,y)|\leqslant
Cr^{\gamma-1}\to 0$ as $r\downarrow 0$. But using A.13 with $j=1$ and A.11 we
have $|D_{r}(r^{\gamma}D_{r}u^{(k)})|\leqslant Cr^{\gamma}$ and hence by
integrating with respect to $r$ and using the above fact that
$r^{\gamma}D_{r}u^{(k)}(r,y)\leqslant Cr^{\gamma-1}\to 0$ as $r\downarrow 0$,
we then have

A.14 $|D_{r}u^{(k)}(r,y)|\leqslant Cr\text{ for small enough $r$ and all
$|y|\leqslant\rho$, $\rho<{\textstyle\frac{1}{2}}$},$

with $C$ depending on $k$, $\gamma$, $\rho$ and $\ell$, and in particular, for
each $\rho<{\textstyle\frac{1}{2}}$,

A.15 $D_{r}u^{(k)}(0_{+},y)=0=\lim_{r\downarrow 0}D_{r}u^{(k)}(r,y)\text{
uniformly for $|y|\leqslant\rho$.}$

We now claim the following polynomial approximation property: For each $u$ as
in A.1, A.2, each $j,k=1,2,\ldots$ with $k\leqslant j$, and each
$(r,y)\in\breve{B}^{+}_{\rho}$, $\rho<{\textstyle\frac{1}{2}}$,

A.16
$\bigl{|}L_{r}^{j-k}u(r,y)-{\textstyle\sum}_{i=1}^{k}c_{ijk}L_{r}^{j-i}u(0,y)r^{2(k-i)}/(2(k-i))!\bigr{|}\leqslant
r^{2k}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|/(2k)!$

where $0<c_{ijk}\leqslant 1$. To prove the case $k=1$, by virtue of A.14,
A.15, we can simply integrate from $0$ to $r$, using
$D_{r}(r^{\gamma}D_{r}L_{r}^{j-1}u(r,y))=r^{\gamma}L^{j}u(r,y)$, followed by a
cancellation of $r^{\gamma}$ from each side of the resulting identity. This
gives

$|D_{r}L_{r}^{j-1}u(r,y)|\leqslant
r\,{\sup}_{B_{\rho}^{+}}|L^{j}u(r,y)|/(\gamma+1),$

and then a second integration using A.15 gives

$\bigl{|}L_{r}^{j-1}u(r,y)-L_{r}^{j-1}u(0,y)\bigr{|}\leqslant(2(\gamma+1))^{-1}r^{2}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u(r,y)|,\,\,j=1,2,\ldots,$

which establishes the case $k=1$. Assume $k+1\leqslant j$ and that A.16 is
correct for $k$. Multiplying each side of A.16 by $r^{\gamma}$ and
integrating, we obtain

$\displaystyle\bigl{|}r^{\gamma}D_{r}L_{r}^{j-k-1}u(r,y)-{\textstyle\sum}_{i=1}^{k}(2(k-i)+\gamma+1)^{-1}c_{ijk}L_{r}^{j-i}u(0,y)\frac{r^{2(k-i)+\gamma+1}}{(2(k-i))!}\bigr{|}$
$\displaystyle\hskip
130.08621pt\leqslant(2k+\gamma+1)^{-1}\frac{r^{2k+\gamma+1}}{(2k)!}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|,$

where we used the fact that
$r^{\gamma}L_{r}^{j-k}u(r,y)=D_{r}(r^{\gamma}D_{r}L_{r}^{j-k-1}u(r,y))$.

After cancelling the factor $r^{\gamma}$ and integrating again, we obtain

$\displaystyle\bigl{|}L_{r}^{j-k-1}u(r,y)-L_{r}^{j-k-1}u(0,y)$
$\displaystyle\hskip
7.22743pt-{\textstyle\sum}_{i=1}^{k}(2(k-i)+\gamma+1)^{-1}(2(k-i)+2)^{-1}c_{ijk}L_{r}^{j-i}u(0,y)\frac{r^{2(k-i)+2}}{(2(k-i))!}\bigr{|}$
$\displaystyle\hskip
10.84006pt\leqslant(2k+\gamma+1)^{-1}(2k+2)^{-1}r^{2k+2}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|/(2k)!\leqslant{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|r^{2k+2}/(2k+2)!$

which confirms the validity of A.16 with $k+1$ in place of $k$. So A.16 is
proved for all $k\leqslant j$, and in particular with $k=j$ and suitable
constants $c_{ij}\in(0,1]$ we get

$u(r,y)={\textstyle\sum}_{i=0}^{j-1}c_{ij}L_{r}^{i}u(0,y)r^{2i}/(2i)!+E_{j}(r,y),\text{where\,}|E_{j}(r,y)|\leqslant
r^{2j}{\sup}_{B_{\rho}^{+}}|L_{r}^{j}u|/(2j)!$

By A.13 and A.11,

${\sup}_{B_{\rho/2}^{+}}|L_{r}^{i}u|/(2i)!={\sup}_{B_{\rho/2}^{+}}|\Delta_{y}^{i}u|/(2i)!\leqslant
C^{i}{\sup}_{B_{\rho/2}^{+}}|u^{(2i)}|/(2i)!\leqslant
C^{i}\bigl{(}\int_{\breve{B}^{+}_{\rho}}|u|^{2}\,d\mu_{+}\bigr{)}^{1/2},$

with $C=C(\gamma,\ell,\rho)$, so, for suitable $C=C(\gamma,\ell)$, we conclude
that $u(r,y)$ has a power series expansion in terms $r^{2}$:

A.17 $u(r,y)={\textstyle\sum}_{j=0}^{\infty}\,\,\,\,a_{j}(y)r^{2j},\quad
0\leqslant r<\sigma,$

where $\sigma=\sigma(\gamma,\ell)\in(0,{\textstyle\frac{1}{2}}]$, and $a_{j}$
satisfies the bounds

${\sup}_{B^{+}_{\sigma}}|a_{j}|\leqslant
C^{j}\bigl{(}\int_{B_{1/2}^{+}}u^{2}\,d\mu_{+}\bigr{)}^{1/2},$

where $C=C(\gamma,\ell)$. Thus A.17 implies

A.18 ${\sup}_{B_{\sigma/2}^{+}}|D_{r}^{j}u(r,y)|\leqslant
C^{j}j!\,\bigl{(}\int_{B_{1/2}^{+}}u^{2}\,d\mu_{+}\bigr{)}^{1/2},\,\,\,C=C(\gamma,\ell).$

Since the same holds with $u^{(k)}$ in place of $u$, and since

$\int_{B_{1/2}^{+}}|u^{(k)}|^{2}\,d\mu_{+}\leqslant
C^{k}(k!)^{2}\int_{B_{3/4}^{+}}u^{2}\,d\mu_{+},$

by A.11, we deduce from A.18 that for suitable
$\sigma=\sigma(\gamma,\ell)\in(0,{\textstyle\frac{1}{2}})$

A.19 ${\sup}_{B_{\sigma}^{+}}|D_{r}^{j}D_{y}^{k}u(r,y)|\leqslant
C^{j+k}j!k!\,\bigl{(}\int_{B_{3/4}^{+}}u^{2}\,d\mu_{+}\bigr{)}^{1/2},$

where $C=C(\gamma,\ell)$, and hence in particular $u$ is real-analytic in the
variables $r^{2}$ and $y_{1},\ldots,y_{\ell}$ in a neighborhood of $(0,0)$ as
claimed.

Finally we show that if $u$ is $\beta$-harmonic in $\breve{B}_{1}^{+}$ then
the power series for $u$ converges in $B_{\rho}^{+}$ for each $\rho<1$, and
also that the homogeneous $\beta$-harmonic polynomials restricted to
$S_{+}^{\ell}$ are complete in $L^{2}(\nu_{+})$ on $S_{+}^{\ell}$, where
$\nu_{+}$ is the measure $d\nu_{+}=\omega_{1}^{\gamma}d\mu_{\ell}$ on
$S_{+}^{\ell}$.

So let $u\in L^{2}(\mu_{+})$ satisfy A.1 and A.2. The above discussion shows
that for suitably small $\sigma$ we can write

A.20 $u={\textstyle\sum}_{j=0}^{\infty}u_{j}\,\,\,\text{ in }B_{\sigma}^{+},$

where $u_{j}$ consists of the homogeneous degree $j$ terms in the power series
expansion of $u$ in $B_{\sigma}$ (and $u_{j}=0$ if there are no such terms).
Then each $u_{j}\neq 0$ is a homogeneous degree j $\beta$-harmonic polynomial
and we let

$\tilde{u}_{j}(\omega)=\rho^{-j}u_{j}(\rho\omega),\,\,\,\hat{u}_{j}(\omega)=\|\tilde{u}_{j}\|_{L^{2}(\nu_{+})}^{-1}\tilde{u}_{j}(\omega),\,\,\,\omega\in\varmathbb{S}^{\ell}_{+},$

and we set $\hat{u}_{j}(\omega)=0$ if $u_{j}=0$. Then, with
$\langle\,,\,\rangle=$ the $L^{2}(\nu_{+})$ inner product, $\langle
u,\hat{u}_{j}\rangle\hat{u}_{j}=u_{j}$ in $B^{+}_{\sigma}$ for each $j$, and
hence by A.20 the series $\sum_{j}\langle u,\hat{u}_{j}\rangle\hat{u}_{j}$
converges smoothly (and also in $L^{2}(\mu_{+})$) to $u$ in $B^{+}_{\sigma}$.

By definition $\rho^{j}\hat{u}_{j}(\omega)$ is either zero or a homogeneous
degree $j$ harmonic polynomial, so by 3.14 ${\rm
div}_{\varmathbb{S}^{\ell}_{+}}(\omega_{1}^{\gamma}\nabla_{\varmathbb{S}^{\ell}_{+}}\hat{u}_{j})=-j(j+\ell+\beta)\omega_{1}^{\gamma}\hat{u}_{j}$,
and hence using the formula 3.13 we can directly check that $\langle
u,\hat{u}_{j}\rangle\hat{u}_{j}$ is $\beta$-harmonic on all of
$\breve{B}_{1}^{+}$. Since by construction it is equal to $u_{j}$ on
$B_{\sigma}$, by unique continuation (applicable since $u$ is real-analytic on
$\breve{B}_{1}^{+}$) we conclude

A.21 $\displaystyle\langle u,\hat{u}_{j}\rangle\hat{u}_{j}\text{ is either
zero or the homogeneous degree $j$ $\beta$-harmonic}$ $\displaystyle\hskip
101.17755pt\text{ polynomial $u_{j}$ on all of $B_{1}^{+}\setminus
S^{\ell}_{+}$ for each $j=0,1,\ldots$}.$

Also, by the orthogonality 3.16,

$\displaystyle\bigl{\|}\sum_{j=p}^{q}\bigl{\langle}u,\hat{u}_{j}\bigr{\rangle}\hat{u}_{j}\bigr{\|}^{2}_{L^{2}(\mu_{+}^{\rho})}$
$\displaystyle=\sum_{j=p}^{q}\int_{0}^{\rho}\bigl{\langle}u(\tau\omega),\hat{u}_{j}(\omega)\bigr{\rangle}^{2}\,\tau^{\gamma+\ell}d\tau$
$\displaystyle\leqslant\int_{0}^{\rho}\bigl{\|}u(\tau\omega)\bigr{\|}^{2}_{L^{2}(\nu_{+})}\,\tau^{\gamma+\ell}d\rho=\bigl{\|}u\bigr{\|}^{2}_{L^{2}(\mu^{\rho}_{+})}\,\,(<\,\infty)$

for each $\rho<1$ and each $p<q$, where $\smash{\mu_{+}^{\rho}}$ is the
measure $\mu_{+}$ on $B_{\rho}$. So $\sum_{j=0}^{q}\langle
u,\hat{u}_{j}\rangle\hat{u}_{j}$ is Cauchy, hence convergent, in
$L^{2}(\mu_{+}^{\rho})$ to a $\beta$-harmonic function $v$ on
$\breve{B}_{\rho}^{+}$. But $v=u$ on $B_{\sigma}^{+}$ and hence, again using
unique continuation, $v=u$ in all of $\breve{B}_{\rho}^{+}$. Thus
$\sum_{j=0}^{q}\langle u,\hat{u}_{j}\rangle\hat{u}_{j}$ converges to $u$ in
$L^{2}(\smash{\mu_{+}^{\rho}})$ for each $\rho<1$ and the convergence is in
$L^{2}(\mu_{+})$ if $\|u\|_{L^{2}(\mu_{+})}<\infty$.

Now observe that the bounds A.19 were established for balls centred at
$(0,0)$, but with only notational changes the same argument gives similar
bounds in balls centred at $(0,y_{0})$ with $|y_{0}|<1$. Specifically for each
$\rho\in(0,1)$ and each $|y_{0}|<\rho$ there is
$\sigma=\sigma(\gamma,\ell,\rho)<{\textstyle\frac{1}{2}}(1-\rho)$ such that

${\sup}_{B_{\sigma}^{+}(0,y_{0})}|D_{r}^{j}D_{y}^{k}u|\leqslant
C^{j+k}j!k!\bigl{(}\int_{B_{(1-\rho)/2}^{+}(0,y_{0})}u^{2}\,d\mu_{+}\bigr{)}^{1/2},\,\,C=C(\gamma,\ell,\rho).$

So in fact, with $\sigma=\sigma(\gamma,\ell,\rho)$ small enough,

${\sup}_{\\{(r,y):r\in[0,\sigma],|y|\leqslant\rho\\}}|D_{r}^{j}D_{y}^{k}u(r,y)|\leqslant
C^{j+k}j!k!\bigl{(}\int_{B_{1}^{+}}u^{2}\,d\mu_{+}\bigr{)}^{1/2},\quad\rho<1.$

Also in $B^{+}_{\rho}\setminus([0,\sigma]\times\varmathbb{R}^{\ell})$ we can
use standard elliptic estimates, so in fact we have

A.22 ${\sup}_{B^{+}_{\rho}}|D_{r}^{j}D_{y}^{k}u(r,y)|\leqslant
C\bigl{(}\int_{B_{1}^{+}}u^{2}\,d\mu_{+}\bigr{)}^{1/2},$

with $C=C(j,k,\gamma,\rho,\ell)$, so the $L^{2}$ convergence of the series
$\sum_{j}\langle u,\hat{u}_{j}\rangle\hat{u}_{j}(=\sum_{j}u_{j})$ proved above
is also $C^{k}$ convergence in $B_{\rho}^{+}$ for each $k\geqslant 1$ and each
$\rho<1$.

Finally to prove the completeness of the homogeneous $\beta$-harmonic
polynomials in $L^{2}(\nu_{+})$ (on $\varmathbb{S}^{\ell}_{+}$), let $\varphi$
be any smooth function on $\varmathbb{S}^{\ell}_{+}$ with $\varphi$ zero in
some neighborhood of $r=0$. By minimizing the energy
$\int_{B_{1}^{+}}(u_{r}^{2}+|u_{y}|^{2})\,r^{\gamma}d\mu$ among functions with
trace $\varphi$ on $\varmathbb{S}_{\ell}$ we obtain a solution of A.4 with
trace $\varphi$ on $\varmathbb{S}^{\ell}_{+}$. The above discussion plus
elliptic boundary regularity shows that $u$ is $C^{0}$ on all of $B_{1}^{+}$
and that the sequence $\\{\sum_{j=0}^{q}u_{j}\\}_{q=0,1,2,\ldots}$, which we
showed above to be convergent to $u$ in $L^{2}(\mu_{+})$ on $B_{1}^{+}$, is
also uniformly convergent to $u$ on all of $B_{1}^{+}$. Hence
$\varphi(\omega)=\sum_{j=0}^{\infty}u_{j}(\omega)$ on
$\varmathbb{S}^{\ell}_{+}$ with the convergence uniform and hence in
$L^{2}(\nu_{+})$. Thus $\varphi$ is represented as an $L^{2}(\nu_{+})$
convergent series of $\beta$-harmonic polynomials on
$\varmathbb{S}^{\ell}_{+}$. Since such $\varphi$ are dense in
$L^{2}(\nu_{+})$, the required completeness is established.

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