# Curvature bound for $L_{p}$ Minkowski problem

Kyeongsu Choi  School of Mathematics, Korea Institute for Advanced Study, 85
Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea<EMAIL_ADDRESS>,
Minhyun Kim  Department of Mathematics, Hanyang University, 04763 Seoul,
Republic of Korea<EMAIL_ADDRESS>and  Taehun Lee  School of
Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu,
Seoul 02455, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We establish curvature estimates for anisotropic Gauss curvature flows. By
using this, we show that given a measure $\mu$ with a positive smooth density
$f$, any solution to the $L_{p}$ Minkowski problem in $\mathbb{R}^{n+1}$ with
$p\leq-n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result
because for each $p\in[-n+2,1)$ there exists a convex hypersurface of class
$C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_{p}$ Minkowski problem
for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is
optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem
in $\mathbb{R}^{3}$.

###### Key words and phrases:

Minkowski problem, curvature flow, regularity estimates

###### 2020 Mathematics Subject Classification:

Primary: 53E99, Secondary: 35B65, 35C06, 35K96, 53A05

##  1\. Introduction

The Minkowski problem asks if there exists a convex body whose surface area
measure is a prescribed measure on $\mathbb{S}^{n}$. This problem was first
proposed and solved by Minkowski [61] for special cases, and later extended
independently by Aleksandrov [1] and Fenchel–Jessen [37] to general measures.
The regularity of solutions to the Minkowski problem has been established by
Pogorelov [62, 63], Cheng–Yau [27], and Caffarelli [15, 16, 17, 18] from the
viewpoint of differential geometry and partial differential equations.

The study of the $L_{p}$ Minkowski problem was initiated by Lutwak [59] as an
important variant of the Minkowski problem. The $L_{p}$ Minkowski problem asks
whether a given measure $\mu$ on $\mathbb{S}^{n}$ arises as the $L_{p}$
surface area measure $S_{p}$ of a convex body $\Omega$, i.e.,

$S_{p}(\Omega,\cdot)=\mu,$ (1.1)

see Section 2.1 for details. Besides the classical Minkowski problem ($p=1$),
there are some significant cases such as the logarithmic Minkowski problem
($p=0$) that characterizes the cone volume, and the centro-affine Minkowski
problem ($p=-n-1$) that is invariant under affine transformations. The
existence of solutions have been extensively studied over the last decades in
various contexts, see [49, 59, 60] for $p>1$, [6, 7, 11, 47, 54, 75] for
$p\in(0,1)$, [8, 10, 24, 26, 73] for $p=0$, and [7, 45, 58, 74, 76] for $p<0$.
Most of the aforementioned references also concern the uniqueness problem. We
refer the readers to [9, 12, 22, 23, 30, 65, 66, 71] for further uniqueness
results and [25, 26, 48, 52, 72] for non-uniqueness results.

In this work, we focus on the regularity of solutions when the prescribed
measure on $\mathbb{S}^{n}$ has a density $f$, namely
$\mathrm{d}\mu=f\,\mathrm{d}\sigma$, where $\sigma$ is the spherical Lebesgue
measure. In this case, the $L_{p}$ Minkowski problem can be interpreted as a
Monge–Ampére type equation

$\displaystyle\det(u_{ij}+u\delta_{ij})=fu^{p-1}\quad\text{on }\mathbb{S}^{n}$
(1.2)

in terms of the support function $u$ of a convex body, provided that $u$ is
positive. It is worth noting that the support function of a convex body is
positive if and only if the origin lies in the interior of a convex body, in
which case the standard regularity theory by Caffarelli [16] can be applied to
(1.2). Therefore, in order to study the regularity, we need to focus on cases
where the origin lies on the boundary, that is, when the support function of a
convex body is nonnegative.

If $u$ is nonnegative, the equation (1.2) is understood in the generalized
sense (of Aleksandrov). We say that the support function $u\in
C(\mathbb{S}^{n})$ of a convex body $\Omega$ or its convex hypersurface
$\Sigma=\partial\Omega$ is called a generalized solution of (1.2) if $\Omega$
satisfies (1.1), namely,

$\int_{E}\mathrm{d}S_{p}(\Omega,z)=\int_{E}f\,\mathrm{d}\sigma\quad\text{for
all Borel sets }E\subset\mathbb{S}^{n},$ (1.3)

where $\sigma$ is the spherical Lebesgue measure.

Let us briefly review the regularity results for the $L_{p}$ Minkowski problem
in the literature. When $p\in(-n-1,-n+1]\cup[n+1,\infty)$, Chou and Wang
proved [33] that solutions to (1.2) have the origin in their interior, and
therefore their support functions are of class $C^{2,\gamma}(\mathbb{S}^{n})$,
provided that $f\in C^{\gamma}(\mathbb{S}^{n})$ is positive. In particular,
solutions to (1.2) are smooth if $f$ is smooth and positive. For the case
$p\in(-\infty,-n-1]$, the smoothness of the solutions can be deduced from the
result of Andrews [3]. We note that, for the affine critical case where
$p=-n-1$ (in which the equation is invariant under affine transformations),
solutions might not exist in general, as described in [33].

However, solutions are not necessarily smooth when $p\in(-n+1,1)\cup(1,n+1)$.
As indicated above, if the origin lies on the boundary of convex bodies, then
one cannot apply the standard regularity theory by Caffarelli, which may allow
non-smooth solutions. Indeed, the $L_{p}$ Minkowski problem, even with
sufficiently regular $f$ could have a solution with the origin on its
boundary. See [11, Example 1.6] for $p\in(-n+1,1)$ and [49, Example 4.1] for
$p\in(1,n+1)$. In these examples, the hypersurfaces are of class
$C^{k,\gamma}$ with $\gamma\in(0,1]$ for $k+\gamma=\frac{2n}{n+p-1}$. Here,

a complete hypersurface $\Sigma$ embedded in $\mathbb{R}^{n+1}$ is said to be
of class $C^{k,\gamma}$,

if each $p\in\Sigma$ has a neighborhood $U$ such that $\Sigma\cap U$ is a
rotated graph of a $C^{k,\gamma}$-function. See also [6, Example 4.2] for
examples of convex bodies with a flat side although the associated $f$ is not
smooth.

We summarize the aforementioned known results in terms of $C^{1,1}$ regularity
of hypersurfaces, which is equivalent to the boundedness of principal
curvatures. For $p\in(-\infty,-n+1]\cup\\{1\\}\cup[n+1,\infty)$, solutions are
smooth if $f$ is smooth. However, for $p\in(1,n+1)$, there exist solutions
that are not $C^{1,1}$ even though $f$ is smooth. For the remaining range,
$p\in(-n+1,1)$, it was not known whether the solutions are $C^{1,1}$ or not
when $f$ is smooth.

In this paper, we completely investigate the $C^{1,1}$ regularity for the
$L_{p}$ Minkowski problem with $p\in(-n+1,1)$ when $f$ is smooth. More
precisely, we show that any generalized solution to (1.2) is a hypersurface of
class $C^{1,1}$ for $p\in(-n+1,-n+2]$; however, there exist solutions that are
not $C^{1,1}$ when $p\in(-n+2,1)$.

######  Theorem 1.1.

Given $p\in(-\infty,-n+2]$ and a positive and smooth function $f$, any
generalized solution $\Sigma$ to (1.2) is a hypersurface of class $C^{1,1}$.
Moreover, its principal curvatures are uniformly bounded by a constant
depending only on $n$, $p$, the diameter of $\Sigma$,
$\|f\|_{C^{2}(\mathbb{S}^{n})}$, and $\min_{\mathbb{S}^{n}}f$.

This is a sharp result due to the following theorem.

######  Theorem 1.2.

Let $p\in[-n+2,1)\cup(1,n+1)$.

  1. (i)

If $p\in[-n+2,1)$, then there exists a generalized solution $\Sigma$ to (1.2)
such that $\Sigma$ is a hypersurface of at most class $C^{1,\frac{1}{n+p-1}}$
and $f$ is a positive smooth function.

  2. (ii)

If $p\in(1,n+1)$, then there exists a generalized solution $\Sigma$ to (1.2)
such that $\Sigma$ is a hypersurface of at most class
$C^{1,\frac{n-p+1}{n+p-1}}$ and $f$ is a positive smooth function.

######  Remark 1.3.

  1. (i)

By combining the known results with Theorem 1.1 and Theorem 1.2, we conclude
the following: the $L_{p}$ Minkowski problem enjoys $C^{1,1}$ regularity of
hypersurfaces if and only if $p\in(-\infty,-n+2]\cup\\{1\\}\cup[n+1,\infty)$.

  2. (ii)

The $C^{1,1}$ regularity is optimal for the case of $p=-n+2$, which includes
the logarithmic Minkowski problem in ${\mathbb{R}}^{3}$ ($n=2$, $p=0$) as a
special case.

  3. (iii)

Chen–Feng–Liu [22] established a uniform diameter estimate for the logarithmic
Minkowski problem in ${\mathbb{R}}^{3}$. Thus, the curvature bound in Theorem
1.1 depends only on $\|f\|_{C^{2}(\mathbb{S}^{n})}$ and
$\min_{\mathbb{S}^{n}}f$ in this case.

On the other hand, in the case $-n-1<p<0$, there is no uniform diameter
estimate depending on $f$ by the result in Jian–Lu–Wang [52].

  4. (iv)

Although the smoothness of solutions and curvature estimates are already known
for $p\in(-\infty,-n+1]$ in Andrews [3] and Chou–Wang [33], we included this
range of $p$ in Theorem 1.1 because our curvature estimates also work in this
range and the proofs are new.

  5. (v)

It was previously known in Chou–Wang [33] that when $p\in(-n+1,1)$, the
associated convex hypersurface is of class $C^{1}$ (or $C^{1,\gamma}$,
respectively) provided that $f\in L^{\infty}(\mathbb{S}^{n})$ (or
$C^{0,1}(\mathbb{S}^{n})$, respectively).

To prove Theorem 1.1, we utilize the anisotropic $\alpha$-Gauss curvature
flow, defined in [3], with $\alpha=\frac{1}{1-p}$. We say that a one-parameter
family of complete convex hypersurfaces $\\{M_{t}\\}_{t\in I}$ is an
anisotropic $\alpha$-Gauss curvature flow if

$\partial_{t}x=-f(\nu)K^{\alpha}\nu,$ (1.4)

holds $\nu$ and $K$ are the unit outward normal and the Gauss curvature of
$M_{t}$ at $x$, respectively. We study local behaviors of the flow (1.4) when
$p\leq-n+2$, or equivalently $\alpha\in(0,\frac{1}{n-1}]$.

######  Theorem 1.4.

Let $\alpha\in(0,\frac{1}{n-1}]$, and let $M_{0}$ be a smooth, closed, and
strictly convex hypersurface in $\mathbb{R}^{n+1}$. Suppose $f$ is a positive
smooth function on $\mathbb{S}^{n}$. If $\\{M_{t}\\}_{0\leq t\leq T}$ is the
anisotropic Gauss curvature flow (1.4), then there exists a positive constant
$C$ depending only on $n$, $\alpha$, the diameter of $M_{0}$, the inradius of
$M_{T}$, $\|f\|_{C^{2}(\mathbb{S}^{n})}$, and $\min_{\mathbb{S}^{n}}f$ such
that

$\displaystyle\max_{i=1,\cdots,n}\lambda_{i}(\cdot,t)\leq
C(1+t^{-\frac{2+\alpha}{(n\alpha+1)\alpha}})\quad\text{for all}\quad 0<t\leq
T.$ (1.5)

The flow approach has been applied to resolve various Minkowski problems,
primarily focusing on asymptotic behavior. For instance, the classical
Minkowski problem was studied using logarithmic Gauss curvature flow in [32,
70]. Moreover, the $L_{p}$ Minkowski problem was investigated in [14, 45, 51];
the dual Minkowski problem, an important variant of the logarithmic Minkowski
problem, was explored in [57]; and the $L_{p}$ dual Minkowski problem was
researched in [19, 20, 46]. The Christoffel-Minkowski problem, which extends
the Gauss curvature in the Minkowski problem to $\sigma_{k}$ curvature, was
examined in [13]. For additional studies related to the $L_{p}$ Minkowski
problem with $f=1$, we refer the readers to [2, 4, 5, 12, 35, 38, 43, 68] for
contraction cases, to [39, 40, 50, 56, 64, 69] for expansion cases, and to
[67] for both types of flows.

It is worth noting that all flows in the aforementioned references, after
normalization, have uniform positive lower bounds for their support functions,
which makes limiting solitons smooth. However, in our range of $p$ with non-
symmetric function $f$, the origin may lie on the limiting soliton, and
therefore (1.2) becomes a degenerate Monge–Ampère type equation and may not
have $C^{2}$ solutions.

Finding $C^{1,1}$ solutions has attracted much attention as partial regularity
theorems of degenerate Monge–Ampère type equations. For the classical
degenerate Monge–Ampère equation, $C^{1,1}$ solutions are found in [41, 44].
Similar results for Minkowski type problem can be found in [21, 42]. For the
flows with degeneracy, we refer readers to [28, 29, 31, 36, 53] for surfaces
with flat sides and [55] for surfaces with an obstacle.

The paper is organized as follows. In Section 2, we provide a brief overview
of the $L_{p}$ Minkowski problem and the anisotropic $\alpha$-Gauss curvature
flow. We also introduce the concept of viscosity solutions of (1.4) and define
some notations. In Section 3, we present evolution equations of various
geometric quantities under the flow. In Section 4, we prove Theorem 1.4, which
provides a uniform upper bound for the principal curvatures of the flow. We
extend the estimate in Theorem 1.4 to the viscosity solution of (1.4) and
present the proof of Theorem 1.1 in Section 5. The proof of Theorem 1.2 is
provided in Section 6.

Acknowledgments. KC has been supported by KIAS Individual Grant MG078901 and
Posco Science Fellowship. TL has been supported by National Research
Foundation of Korea grant RS-2023-00211258.

##  2\. Preliminaries

In this section, we recall precise definitions for the $L_{p}$ Minkowski
problem and the anisotropic $\alpha$-Gauss curvature flow, which are closely
related to each other. We also make some notations.

###  2.1. $L_{p}$ Minkowski problem

The Minkowski problem asks for the construction of a convex body, a compact
convex set in $\mathbb{R}^{n+1}$ with a nonempty interior, whose Gauss
curvature is prescribed. It is equivalent to finding a convex body $\Omega$
satisfying

$S(\Omega,\cdot)=\mu$ (2.1)

for a given measure $\mu$ on $\mathbb{S}^{n}$, where $S(\Omega,\cdot)$ is the
surface area measure of $\Omega$ determined by the Aleksandrov variational
formula

$\left.\frac{\mathrm{d}V(\Omega+t\Omega_{0})}{\mathrm{d}t}\right|_{t=0^{+}}=\int_{\mathbb{S}^{n}}u_{\Omega_{0}}(z)\,\mathrm{d}S(\Omega,z)\quad\text{for
any convex body }\Omega_{0}.$ (2.2)

Here, $V(\Omega)$ denotes the volume of $\Omega$, and
$u_{\Omega_{0}}:\mathbb{S}^{n}\to\mathbb{R}$ is the support function of
$\Omega_{0}$ defined by $u_{\Omega_{0}}(z)=\max\\{z\cdot x:x\in\Omega_{0}\\}$.

To describe the $L_{p}$ Minkowski problem, we recall Firey’s [38] $p$-linear
combination of convex bodies: for $p\geq 1$, $t_{1},t_{2}>0$, and convex
bodies $\Omega_{1}$, $\Omega_{2}$ containing the origin,
$t_{1}\cdot_{p}\Omega_{1}+_{p}t_{2}\cdot_{p}\Omega_{2}$ is defined by the
convex body with support function
$(t_{1}u_{\Omega_{1}}^{p}+t_{2}u_{\Omega_{2}}^{p})^{1/p}$. Then, the $L_{p}$
surface area measure $S_{p}(\Omega,\cdot)$ of a convex body $\Omega$ is
characterized by

$\left.\frac{\mathrm{d}V(\Omega+_{p}t\cdot_{p}\Omega_{0})}{\mathrm{d}t}\right|_{t=0^{+}}=\frac{1}{p}\int_{\mathbb{S}^{n}}u_{\Omega_{0}}^{p}(z)\,\mathrm{d}S_{p}(\Omega,z)\quad\text{for
any convex body }\Omega_{0}.$ (2.3)

It is known [59] that $S_{p}(\Omega,\cdot)$ is related to $S(\Omega,\cdot)$ by
$S_{p}(\Omega,\cdot)=u_{\Omega}^{1-p}S(\Omega,\cdot)$, and this relation makes
it possible to define the $L_{p}$ surface area measure for all
$p\in\mathbb{R}$. The $L_{p}$ Minkowski problem is to find a convex body
$\Omega$ satisfying

$S_{p}(\Omega,\cdot)=\mu$ (2.4)

for a given measure $\mu$ on $\mathbb{S}^{n}$. If $\mu$ has a density $f$ so
that $\mu=f\,\mathrm{d}\sigma$, then (2.4) is equal to (1.2).

###  2.2. Anisotropic $\alpha$-Gauss curvature flow

Let us consider an anisotropic Gauss curvature flow (1.4) for $\alpha>0$ and a
positive smooth function $f$ defined on $\mathbb{S}^{n}$. It follows from the
result of Andrews [3] that any closed convex hypersurface converges to a point
in a finite time $T_{\max}>0$ under the flow (1.4).

######  Theorem A (Andrews [3]).

Let $\alpha>0$ and $f\in C^{\infty}(\mathbb{S}^{n})$. If $M_{0}$ is a smooth,
closed, strictly convex hypersurface in ${\mathbb{R}}^{n+1}$, then there
exists a unique solution $\\{M_{t}\\}_{0<t<T_{\max}}$ to (1.4) starting from
$M_{0}$ such that $M_{t}$ converges to a point as time $t$ approaches its
maximal existence time $T_{\max}<\infty$.

Additionally, Andrews [3] defined viscosity solutions for the flow (1.4) that
starts from general convex bodies. We adopt the definition to include
generalized solutions to the $L_{p}$ Minkowski problems.

######  Definition 2.1 (Viscosity Solution [3]).

Let $\\{\Omega_{t}\\}_{0<t\leq T}$ be a family of convex regions, and let $f$
be a positive smooth function on $\mathbb{S}^{n}$. We say that
$\\{\Omega_{t}\\}_{0<t\leq T}$ is a viscosity solution to the flow (1.4) if it
satisfies the following two conditions:

  1. (i)

For any strictly convex, smooth hypersurface $M_{0}^{-}$ contained in
$\Omega_{0}$, the evolution $M_{t}^{-}$ of $M_{0}^{-}$ under the flow (1.4) is
contained in $\Omega_{t}$ up to the maximal existence time of $M_{t}^{-}$.

  2. (ii)

For any strictly convex, smooth hypersurface $M_{0}^{+}$ enclosing
$\Omega_{0}$, the evolution $M_{t}^{+}$ of $M_{0}^{+}$ under the flow (1.4)
encloses $\Omega_{t}$ for all $0\leq t\leq T$.

We refer to $\\{\partial\Omega_{t}\\}_{0<t\leq T}$ as a viscosity solution as
well, by slight abuse of notation.

The existence and uniqueness of viscosity solutions are well-established in
that paper.

######  Theorem B (Andrews, Theorem 15 in [3]).

Let $\alpha>0$ and $f\in C^{\infty}(\mathbb{S}^{n})$. If $\Omega_{0}$ is an
open bounded convex region, then there exists a unique viscosity solution
$\\{\Omega_{t}\\}_{0<t<T_{\max}}$, which converges to $\Omega_{0}$ in
Hausdorff distance as $t$ approaches zero. Moreover, $\Omega_{t}$ converges to
a point as $t$ approaches its maximal existence time $T_{\max}<\infty$.

Lastly, we recall the upper bound for the Gauss curvature provided in [3]. We
denote by $\rho_{-}(M_{T})$ the inradius of $M_{T}$.

######  Theorem C.

Let $M_{t}$ be a strictly convex and smooth solution to the flow (1.4) for
$t\in[0,T]$. Then there is a positive constant $C$ depending only on $n$,
$\alpha$, $\max_{\mathbb{S}^{n}}f$, $\min_{\mathbb{S}^{n}}f$,
$\operatorname{diam}(M_{0})$, and $\rho_{-}(M_{T})$ such that

$\displaystyle K(\cdot,t)\leq C(1+t^{-\frac{n}{n\alpha+1}})\quad\text{for
all}\quad 0<t\leq T.$ (2.5)

###  2.3. Notations

Let $\Sigma$ be a strictly convex hypersurface in ${\mathbb{R}}^{n+1}$. Then,
we denote by $\\{g_{ij}\\}$ and $\\{h_{ij}\\}$ the induced metric and the
second fundamental form, respectively. The inverse of the metric is denoted by
$g^{ij}$. Since $\Sigma$ is strictly convex, $\\{h_{ij}\\}$ has the inverse
matrix $\\{b^{ij}\\}$, namely

$b^{ik}h_{kj}=\delta_{ij},$ (2.6)

where $\delta_{ij}$ is the Kronecker delta. Also, we denote

$\operatorname{tr}(b)=b^{ij}g_{ij}.$ (2.7)

Given a function $f\in C^{\infty}(\mathbb{S}^{n})$, we may extend its domain
to ${\mathbb{R}}^{n+1}\setminus\\{0\\}\to\mathbb{R}$ so that by abuse of
notation we define $f(z):=f(z/|z|)$ for $z\neq 0$. We denote by $Df$ the
gradient of $f$ with respect to the standard euclidean metric, and denote by
$D_{z}f=\left\langle{Df},{z}\right\rangle$ the directional derivative of $f$
in the direction $z$. For the purpose of brevity, we define

$\displaystyle f_{i}=D_{\nabla_{i}x}f,$ $\displaystyle
f_{ij}=D^{2}_{\nabla_{i}x,\nabla_{j}x}f$ (2.8)

for $\nabla_{i}x,\nabla_{j}x\in T_{x}\Sigma$, and

$\displaystyle f^{i}=g^{ij}f_{j},$ $\displaystyle f^{ij}=g^{ik}g^{ij}f_{kl}.$
(2.9)

######  Remark 2.2.

The indices of $f_{i},f_{ij}$ are determined by local charts of $\Sigma$
rather than by the fixed coordinates of the ambient space $\mathbb{R}^{n+1}$.
Namely, $f_{i}$ would be different from $\frac{\partial}{\partial z_{i}}f(z)$.

On the other hand, we can observe
$|\bar{\nabla}_{\mathbb{S}^{n}}f|^{2}=f_{i}f^{i}$ and
$|\bar{\nabla}_{\mathbb{S}^{n}}^{2}f|^{2}=f_{ij}f^{ij}$, where
$\bar{\nabla}_{\mathbb{S}^{n}}$ is the Levi-Civita connection of the unit
sphere.

######  Proposition 2.3.

The function $f(\nu)$ of the outward point unit normal vector $\nu$ satisfies

$\displaystyle\nabla_{i}f(\nu)=h_{ik}f^{k},$
$\displaystyle\nabla_{i}\nabla_{j}f(\nu)=f^{k}\nabla_{k}h_{ij}+h_{ik}h_{jl}f^{kl}.$
(2.10)

###### Proof.

We directly compute
$\nabla_{i}f=\langle\nabla_{i}\nu,Df\rangle=h_{ij}g^{jk}\langle\nabla_{k}x,Df\rangle=h_{ij}f^{j}$.
We can obtain the second identity in the same manner. ∎

We recall the linearized operator

${\mathcal{L}}=\alpha
fK^{\alpha}b^{ij}\nabla_{i}\nabla_{j}+K^{\alpha}f^{k}\nabla_{k},$ (2.11)

and the associated inner product

$\displaystyle\left\langle{\nabla v},{\nabla w}\right\rangle_{W}=\alpha
fK^{\alpha}b^{ij}\nabla_{i}v\nabla_{j}w$ (2.12)

for $v,w\in C^{1}(\Sigma)$, where

$\displaystyle\nabla v=\nabla_{i}v\nabla^{i}x.$ (2.13)

Furthermore, we denote the support function and the speed of the flow (1.4) by

$\displaystyle u=\langle x,\nu\rangle,$ $\displaystyle F=fK^{\alpha},$ (2.14)

respectively.

##  3\. Evolution equations

In this section we obtain evolution equations for geometric quantities under
the flow (1.4). The proofs are standard, and some of them can be found in [3],
where the calculations are performed in the coordinates of $\mathbb{S}^{n}$
with respect to the parametrization
$x\circ\nu^{-1}:\mathbb{S}^{n}\rightarrow{\mathbb{R}}$ (see also [43] for the
case $f\equiv 1$). We derive equations by using the covariant derivative on
the hypersurface $M_{t}$.

######  Lemma 3.1.

Given a solution to the flow (1.4), the following equations hold:

  1. (i)

$\partial_{t}g_{ij}=-2Fh_{ij}$,

  2. (ii)

$\partial_{t}\nu={\mathcal{L}}\nu+\alpha FH\nu$,

  3. (iii)

$\partial_{t}|x|^{2}={\mathcal{L}}|x|^{2}+2(n\alpha-1)Fu-2\alpha
F\operatorname{tr}(b)-2K^{\alpha}f^{k}\langle x,\nabla_{k}x\rangle$,

  4. (iv)

$\partial_{t}h_{ij}$ $={\mathcal{L}}h_{ij}+(\alpha^{2}b^{pq}b^{kl}-\alpha
b^{kp}b^{lq})F\nabla_{i}h_{pq}\nabla_{j}h_{kl}+h_{ik}f^{k}\nabla_{j}K^{\alpha}+h_{jk}f^{k}\nabla_{i}K^{\alpha}$
$+K^{\alpha}f^{kl}h_{ik}h_{jl}-(n\alpha+1)Fh_{ik}h^{k}_{j}+\alpha FHh_{ij}$,

  5. (v)

$\partial_{t}F={\mathcal{L}}F+\alpha F^{2}H$.

###### Proof.

(i) Since $\left\langle{\partial_{j}x},{\nu}\right\rangle=0$, we have

$\displaystyle\left\langle{\partial_{t}\partial_{i}x},{\partial_{j}x}\right\rangle=\left\langle{\partial_{i}\partial_{t}x},{\partial_{j}x}\right\rangle=\left\langle{\partial_{i}(-F\nu)},{\partial_{j}x}\right\rangle=-Fh_{ij}.$
(3.1)

Similarly, we obtain
$\left\langle{\partial_{i}x},{\partial_{t}\partial_{j}x}\right\rangle=-Fh_{ij}$
and thus

$\displaystyle\partial_{t}g_{ij}=\left\langle{\partial_{t}\partial_{i}x},{\partial_{j}x}\right\rangle+\left\langle{\partial_{i}x},{\partial_{t}\partial_{j}x}\right\rangle=-2Fh_{ij}$
(3.2)

as desired.

(ii) We first compute

$\displaystyle\left\langle{\partial_{t}\nu},{\partial_{k}x}\right\rangle=-\left\langle{\nu},{\partial_{k}\partial_{t}x}\right\rangle=-\left\langle{\nu},{\partial_{k}(-F\nu)}\right\rangle=\partial_{k}F.$
(3.3)

Thus we have

$\partial_{t}\nu=\nabla F.$ (3.4)

On the other hand, observing

$\nabla_{i}\nabla_{j}\nu=\nabla_{i}(h_{jk}\nabla^{k}x)=\nabla_{k}h_{ij}\nabla^{k}x-h_{i}^{k}h_{jk}\nu,$
(3.5)

we have

$\displaystyle\alpha Fb^{ij}\nabla_{i}\nabla_{j}\nu=f\nabla K^{\alpha}-\alpha
FH\nu.$ (3.6)

Also, Proposition 2.3 implies

$K^{\alpha}f^{k}\nabla_{k}\nu=K^{\alpha}f^{k}h_{kl}\nabla^{l}x=K^{\alpha}\langle\nabla_{l}f,\nabla^{l}x\rangle=K^{\alpha}\nabla
f.$ (3.7)

Remembering $F=\alpha K^{\alpha}$ and the definition of ${\mathcal{L}}$, the
equations above give us

$\displaystyle(\partial_{t}-{\mathcal{L}})\nu=\nabla(fK^{\alpha})-f\nabla
K^{\alpha}+\alpha FH\nu-K^{\alpha}\nabla f=\alpha FH\nu.$ (3.8)

(iii) Note that

$\displaystyle\partial_{t}|x|^{2}$
$\displaystyle=2\left\langle{x},{x_{t}}\right\rangle=-2Fu,$ (3.9)
$\displaystyle\nabla_{i}\nabla_{j}|x|^{2}$
$\displaystyle=2\left\langle{\nabla_{i}x},{\nabla_{j}x}\right\rangle+2\left\langle{x},{\nabla_{i}\nabla_{j}x}\right\rangle=2g_{ij}-2h_{ij}u,$
(3.10) $\displaystyle\alpha Fb^{ij}\nabla_{i}\nabla_{j}|x|^{2}$
$\displaystyle=2\alpha F\operatorname{tr}(b)-2n\alpha Fu,$ (3.11)
$\displaystyle K^{\alpha}f^{k}\nabla_{k}|x|^{2}$
$\displaystyle=2K^{\alpha}f^{k}\langle x,D_{k}x\rangle.$ (3.12)

Thus

$\displaystyle(\partial_{t}-{\mathcal{L}})|x|^{2}=-2Fu-2\alpha
F\operatorname{tr}(b)+2n\alpha Fu-2K^{\alpha}D_{x}f.$ (3.13)

(iv) We note that

$\displaystyle\nabla_{j}\nabla
F=\nabla_{j}(\nabla^{k}F\nabla_{k}x)=\nabla_{j}\nabla^{k}F\nabla_{k}x-(\nabla^{k}F)h_{kj}\nu,$
(3.14)

and thus remembering $\nu_{t}=\nabla F$ in (3.4) we have

$\displaystyle\partial_{t}h_{ij}$
$\displaystyle=\left\langle{\nabla_{i}\partial_{t}x},{\nabla_{j}\nu}\right\rangle+\left\langle{\nabla_{i}x},{\nabla_{j}\partial_{t}\nu}\right\rangle=\left\langle{\nabla_{i}(-F\nu)},{\nabla_{j}\nu}\right\rangle+\left\langle{\nabla_{i}x},{\nabla_{j}\nabla
F}\right\rangle$ (3.15)
$\displaystyle=-Fh_{ik}h^{k}_{j}+\nabla_{i}\nabla_{j}F.$ (3.16)

Using the Gauss equation $R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}$ and the Codazzi
equation, we obtain

$\displaystyle\nabla_{k}\nabla_{l}h_{ij}$
$\displaystyle=\nabla_{k}\nabla_{i}h_{jl}=\nabla_{i}\nabla_{k}h_{jl}+R_{kijm}h^{m}_{l}+R_{kilm}h_{j}^{m}$
(3.17)
$\displaystyle=\nabla_{i}\nabla_{j}h_{kl}+(h_{kj}h_{im}-h_{km}h_{ij})h^{m}_{l}+(h_{kl}h_{im}-h_{km}h_{il})h_{j}^{m},$
(3.18)

and therefore

$\displaystyle\alpha Fb^{kl}\nabla_{k}\nabla_{l}h_{ij}=\alpha
Fb^{kl}\nabla_{i}\nabla_{j}h_{kl}+\alpha F(-Hh_{ij}+nh_{im}h^{m}_{j}).$ (3.19)

Hence

$\displaystyle\partial_{t}h_{ij}$ $\displaystyle=\alpha
Fb^{kl}\nabla_{k}\nabla_{l}h_{ij}-(n\alpha+1)Fh_{ik}h^{k}_{j}+\nabla_{i}\nabla_{j}F-\alpha
Fb^{kl}\nabla_{i}\nabla_{j}h_{kl}+\alpha FHh_{ij}.$ (3.20)

By using $\nabla_{j}K^{\alpha}=\alpha K^{\alpha}b^{kl}\nabla_{j}h_{kl}$ and

$\displaystyle\nabla_{i}\nabla_{j}K^{\alpha}=\alpha
K^{\alpha}b^{kl}\nabla_{i}\nabla_{j}h_{kl}+\alpha\nabla_{i}(K^{\alpha}b^{kl})\nabla_{j}h_{kl},$
(3.21)

we obtain

$\displaystyle\nabla_{i}\nabla_{j}F-\alpha Fb^{kl}\nabla_{i}\nabla_{j}h_{kl}$
$\displaystyle=K^{\alpha}\nabla_{i}\nabla_{j}f+\nabla_{i}f\nabla_{j}K^{\alpha}+\nabla_{j}f\nabla_{i}K^{\alpha}$
(3.22) $\displaystyle\quad+\alpha
f\nabla_{i}(K^{\alpha}b^{kl})\nabla_{j}h_{kl}.$ (3.23)

Thus, combining above equations yields

$\displaystyle\partial_{t}h_{ij}$ $\displaystyle=\alpha
Fb^{kl}\nabla_{k}\nabla_{l}h_{ij}+(\alpha^{2}b^{pq}b^{kl}-\alpha
b^{kp}b^{lq})F\nabla_{i}h_{pq}\nabla_{j}h_{kl}+K^{\alpha}\nabla_{i}\nabla_{j}f$
(3.24)
$\displaystyle\quad+\nabla_{i}f\nabla_{j}K^{\alpha}+\nabla_{j}f\nabla_{i}K^{\alpha}-(n\alpha+1)Fh_{ik}h^{k}_{j}+\alpha
FHh_{ij}.$ (3.25)

Then, applying Proposition 2.3 completes the proof.

(v) Since $\left\langle{Df},{\nu}\right\rangle=0$, the equation (ii) implies
$\left\langle{Df},{(\partial_{t}-{\mathcal{L}})\nu}\right\rangle=0$. Thus,
observing

$\displaystyle\alpha Fb^{ij}\nabla_{i}\nu\nabla_{j}\nu=\alpha
Fb^{ij}h_{ik}h_{jl}\nabla^{k}x\nabla^{l}x=\alpha
Fh_{kl}\nabla^{k}x\nabla^{l}x,$ (3.26)

we have

$(\partial_{t}-{\mathcal{L}})f=Df(\partial_{t}-{\mathcal{L}})\nu-\alpha
Fb^{ij}D^{2}f\nabla_{i}\nu\nabla_{j}\nu=-\alpha Ff^{kl}h_{kl}.$ (3.27)

To derive the equation for $K^{\alpha}$, we first check

$\displaystyle\partial_{t}g^{ij}=-g^{ik}g^{jl}\partial_{t}g_{kl}=-2g^{ik}g^{jl}(-Fh_{kl})=2Fh^{ij}.$
(3.28)

Since $K=\det(g^{ik}h_{kj})$, we have

$\displaystyle(\partial_{t}-{\mathcal{L}})K^{\alpha}$ $\displaystyle=\alpha
K^{\alpha}g_{ij}\partial_{t}g^{ij}+\alpha
K^{\alpha}b^{ij}(\partial_{t}-{\mathcal{L}})h_{ij}-\alpha
Fb^{kl}\nabla_{k}(K^{\alpha}b^{ij})\nabla_{l}h_{ij}.$ (3.29)

Using (3.28), the first term becomes

$\displaystyle\alpha K^{\alpha}g_{ij}\partial_{t}g^{ij}=2\alpha K^{\alpha}FH.$
(3.30)

For the other two terms, we see that

$\displaystyle\alpha K^{\alpha}b^{ij}(\alpha^{2}b^{pq}b^{kl}-\alpha
b^{kp}b^{lq})F\nabla_{i}h_{pq}\nabla_{j}h_{kl}$ $\displaystyle=\alpha
Fb^{kl}\nabla_{k}(K^{\alpha}b^{ij})\nabla_{l}h_{ij},$ (3.31)
$\displaystyle\alpha K^{\alpha}b^{ij}(-(n\alpha+1)Fh_{ik}h^{k}_{j}+\alpha
FHh_{ij})$ $\displaystyle=-\alpha K^{\alpha}FH.$ (3.32)

Hence, (3.29) becomes

$\displaystyle\partial_{t}K^{\alpha}$
$\displaystyle={\mathcal{L}}K^{\alpha}+\alpha
K^{\alpha}b^{ij}(2h_{ik}f^{k}\nabla_{j}K^{\alpha}+K^{\alpha}f^{kl}h_{ik}h_{jl})+\alpha
K^{\alpha}FH$ (3.33) $\displaystyle={\mathcal{L}}K^{\alpha}+2\alpha
K^{\alpha}b^{ij}f^{k}\nabla_{k}K^{\alpha}+\alpha
K^{2\alpha}f^{kl}h_{kl}+\alpha K^{\alpha}FH.$ (3.34)

Combining these equations for $f$ and $K^{\alpha}$, we obtain for
$F=fK^{\alpha}$ that

$\displaystyle(\partial_{t}-{\mathcal{L}})F$
$\displaystyle=f(\partial_{t}-{\mathcal{L}})K^{\alpha}+K^{\alpha}(\partial_{t}-{\mathcal{L}})f-2\alpha
Fb^{ij}\nabla_{i}f\nabla_{j}K^{\alpha}$ (3.35) $\displaystyle=2\alpha
Ff^{k}\nabla_{k}K^{\alpha}+\alpha K^{\alpha}Ff^{kl}h_{kl}+\alpha F^{2}H$
(3.36) $\displaystyle\quad-\alpha K^{\alpha}Ff^{kl}h_{kl}-2\alpha
Fb^{ij}h_{ik}f^{k}\nabla_{j}K^{\alpha}$ (3.37) $\displaystyle=\alpha F^{2}H.$
(3.38)

This completes the proof. ∎

##  4\. Principal curvature bound

In this section we provide a uniform upper bound for principal curvatures. The
estimate heavily relies on diameter and the upper bound on the Gauss curvature
$K$ in C. Since the minimum eigenvalue is not necessarily smooth, we need the
following derivatives of a smooth approximation for the minimum eigenvalue.

######  Lemma 4.1.

Let $\mu$ be the multiplicity of the smallest principal curvature at a point
$x_{0}$ on ${M}_{t_{0}}$ for $t_{0}>0$ so that
$\lambda_{1}=\cdots=\lambda_{\mu}<\lambda_{\mu+1}\leq\cdots\leq\lambda_{n}$,
where $\lambda_{1},\cdots,\lambda_{n}$ are the principal curvatures. Suppose
$\varphi$ is a smooth function defined on $\mathcal{M}=\cup_{0<t\leq
t_{0}}M_{t}\times\\{t\\}$ such that $\varphi\leq\lambda_{1}$ on $\mathcal{M}$
and $\varphi=\lambda_{1}$ at $X_{0}=(x_{0},t_{0})$. Then, at the point $X_{0}$
with a chart of $g_{ij}=\delta_{ij}$ and $h_{ij}=\lambda_{i}\delta_{ij}$, we
have $\nabla_{i}h_{kl}=\nabla_{i}\varphi\delta_{kl}$ for $1\leq k,l\leq\mu$,

$\displaystyle\nabla_{i}\nabla_{i}\varphi\leq\nabla_{i}\nabla_{i}h_{11}-\sum_{l>\mu}\frac{2(\nabla_{i}h_{1l})^{2}}{\lambda_{l}-\lambda_{1}},\quad\text{and}\quad\partial_{t}\varphi\geq\partial_{t}h_{11}+2F\lambda_{1}^{2}.$
(4.1)

###### Proof.

The identity for $\nabla\varphi$ and the inequality for $\nabla^{2}\varphi$
are already given in [12]. To prove the inequality for $\partial_{t}\varphi$,
we note that $\lambda_{1}\leq h_{11}/g_{11}$. Indeed, if
$\\{E_{1},\cdots,E_{n}\\}$ is an orthonormal basis of the tangent space
$T(M_{t_{0}})_{x_{0}}$ such that $L(E_{j})=\lambda_{j}E_{j}$ for the
Weingarten map $L$, then it follows by writing $\nabla_{1}x=a_{j}E_{j}$ that

$\displaystyle
h_{11}=\left\langle{L(\nabla_{1}x)},{\nabla_{1}x}\right\rangle=\sum_{j=1}^{n}\lambda_{j}a_{j}^{2}\geq\lambda_{1}\sum_{j=1}^{n}a_{j}^{2}=\lambda_{1}g_{11}.$
(4.2)

Thus, $h_{11}/g_{11}-\varphi\geq 0$ and the minimum occurs at $X_{0}$. Taking
derivative in time, we obtain

$\displaystyle
0\geq\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=t_{0}}\left(\frac{h_{11}}{g_{11}}-\varphi\right)=\frac{\partial_{t}h_{11}}{g_{11}}-\frac{h_{11}\partial_{t}g_{11}}{g_{11}^{2}}-\partial_{t}\varphi.$
(4.3)

Then the conclusion follows from $g_{11}=1$, $h_{11}=\lambda_{1}$, and
$\partial_{t}g_{11}=-2Fh_{11}=-2F\lambda_{1}$. ∎

######  Lemma 4.2.

Let $M_{t}$ be a strictly convex and smooth solution to the flow (1.4) for
$t\in[0,T]$, and let $\varphi$ be the function satisfying the conditions in
Lemma 4.1. Then the following holds at the maximum point of $\varphi$:

$\displaystyle\partial_{t}(F\varphi^{-1})$
$\displaystyle\leq{\mathcal{L}}(F\varphi^{-1})-\frac{\alpha
F^{2}}{\lambda_{1}^{4}}(\nabla_{1}h_{11})^{2}-\sum_{k>\mu}\frac{2\alpha
F^{2}(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})}$ (4.4)
$\displaystyle\quad-\frac{(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+CF^{2},$ (4.5)

where $C$ is a constant depending only on $n$, $\alpha$,
$\|f\|_{C^{2}(\mathbb{S}^{n})}$, and $\min_{\mathbb{S}^{n}}f$,.

###### Proof.

From the evolution equation of $h_{ij}$ in Lemma 3.1, we have

$\displaystyle(\partial_{t}-{\mathcal{L}})h_{11}$
$\displaystyle=(\alpha^{2}b^{pq}b^{kl}-\alpha
b^{kp}b^{lq})F\nabla_{1}h_{pq}\nabla_{1}h_{kl}+2h_{1k}f^{k}\nabla_{1}K^{\alpha}+K^{\alpha}f^{kl}h_{1k}h_{1l}$
(4.6) $\displaystyle\quad-(n\alpha+1)Fh^{k}_{1}h_{k1}+\alpha FHh_{11}.$ (4.7)

By Lemma 4.1, we obtain at the point

$\displaystyle(\partial_{t}-{\mathcal{L}})\varphi$
$\displaystyle\geq(\partial_{t}-{\mathcal{L}})h_{11}+\sum_{k}\sum_{l>\mu}\frac{2\alpha
F(\nabla_{k}h_{1l})^{2}}{\lambda_{k}(\lambda_{l}-\lambda_{1})}+2F\lambda_{1}^{2}$
(4.8) $\displaystyle=(\alpha^{2}b^{pq}b^{kl}-\alpha
b^{kp}b^{lq})F\nabla_{1}h_{pq}\nabla_{1}h_{kl}+\sum_{k}\sum_{l>\mu}\frac{2\alpha
F(\nabla_{k}h_{1l})^{2}}{\lambda_{k}(\lambda_{l}-\lambda_{1})}+2f_{1}\lambda_{1}\nabla_{1}K^{\alpha}$
(4.9)
$\displaystyle\quad+K^{\alpha}f_{11}\lambda_{1}^{2}-(n\alpha-1)F\lambda_{1}^{2}+\alpha
FH\lambda_{1}.$ (4.10)

Since $\nabla_{1}K^{\alpha}=\alpha K^{\alpha}b^{ij}\nabla_{1}h_{ij}$, we have

$\displaystyle\alpha^{2}b^{pq}b^{kl}F\nabla_{1}h_{pq}\nabla_{1}h_{kl}=fK^{-\alpha}(\nabla_{1}K^{\alpha})^{2}.$
(4.11)

On the other hand, by Lemma 4.1 and the Codazzi equation, we get
$\nabla_{1}h_{ik}=0$ for $1<k\leq\mu$, which implies

$\displaystyle\alpha
b^{kp}b^{lq}F\nabla_{1}h_{pq}\nabla_{1}h_{kl}=\frac{\alpha
F(\nabla_{1}h_{11})^{2}}{\lambda_{1}^{2}}+\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}\lambda_{k}}+\sum_{k,l>\mu}\frac{\alpha
F(\nabla_{1}h_{kl})^{2}}{\lambda_{k}\lambda_{l}},$ (4.12)

and

$\displaystyle\sum_{k}\sum_{l>\mu}\frac{2\alpha
F(\nabla_{k}h_{1l})^{2}}{\lambda_{k}(\lambda_{l}-\lambda_{1})}=\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}(\lambda_{k}-\lambda_{1})}+\sum_{k,l>\mu}\frac{2\alpha
F(\nabla_{1}h_{kl})^{2}}{\lambda_{k}(\lambda_{l}-\lambda_{1})}.$ (4.13)

Hence,

$\displaystyle-\alpha
b^{kp}b^{lq}F\nabla_{1}h_{pq}\nabla_{1}h_{kl}+\sum_{k}\sum_{l>\mu}\frac{2\alpha
F(\nabla_{k}h_{1l})^{2}}{\lambda_{k}(\lambda_{l}-\lambda_{1})}$ (4.14)
$\displaystyle\geq-\frac{\alpha
F(\nabla_{1}h_{11})^{2}}{\lambda_{1}^{2}}+\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{k}(\lambda_{k}-\lambda_{1})}.$ (4.15)

Moreover, using Proposition 2.3 and $F=fK^{\alpha}$, we have

$\displaystyle
fK^{-\alpha}(\nabla_{1}K^{\alpha})^{2}+2\lambda_{1}f_{1}\nabla_{1}K^{\alpha}=\frac{(\nabla_{1}F)^{2}}{F}-\frac{K^{\alpha}(\lambda_{1}f_{1})^{2}}{f}=\frac{(\nabla_{1}F)^{2}}{F}-\frac{F\lambda_{1}^{2}f_{1}^{2}}{f^{2}}.$
(4.16)

Then we have

$\displaystyle(\partial_{t}-{\mathcal{L}})\varphi$
$\displaystyle\geq\frac{(\nabla_{1}F)^{2}}{F}-\alpha
F\lambda_{1}^{-2}(\nabla_{1}h_{11})^{2}+\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{k}(\lambda_{k}-\lambda_{1})}$ (4.17)
$\displaystyle\quad+F\lambda_{1}^{2}\left(\frac{f_{11}}{f}-\frac{f_{1}^{2}}{f^{2}}-n\alpha+1\right)+\alpha
FH\lambda_{1}.$ (4.18)

Observing $\nabla_{j}\varphi^{-1}=-\varphi^{-2}\nabla_{j}\varphi$ and
$\nabla_{i}\nabla_{j}\varphi^{-1}=-\varphi^{-2}\nabla_{i}\nabla_{j}\varphi+2\varphi^{-3}\nabla_{i}\varphi\nabla_{j}\varphi$,
we have

$\displaystyle(\partial_{t}-{\mathcal{L}})\varphi^{-1}=-\lambda_{1}^{-2}(\partial_{t}-{\mathcal{L}})\varphi-\sum_{k=1}^{n}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}\lambda_{k}}.$ (4.19)

Since

$\displaystyle-\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{2}\lambda_{k}(\lambda_{k}-\lambda_{1})}-\sum_{k=2}^{n}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}\lambda_{k}}\leq-\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})},$ (4.20)

combining with (4.1) yields

$\displaystyle(\partial_{t}-{\mathcal{L}})\varphi^{-1}$
$\displaystyle\leq-\frac{(\nabla_{1}F)^{2}}{F\lambda_{1}^{2}}-\alpha
F\lambda_{1}^{-4}(\nabla_{1}h_{11})^{2}-\sum_{k>\mu}\frac{2\alpha
F(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})}$ (4.21)
$\displaystyle\quad-F\left(\frac{f_{11}}{f}-\frac{f_{1}^{2}}{f^{2}}-n\alpha+1\right)-\alpha
FH\lambda_{1}^{-1}.$ (4.22)

Hence, by using the evolution equation of $F$ in Lemma 3.1, we have

$\displaystyle\partial_{t}(F\varphi^{-1})$
$\displaystyle={\mathcal{L}}(F\varphi^{-1})+F(\partial_{t}-{\mathcal{L}})\varphi^{-1}+\varphi^{-1}(\partial_{t}-{\mathcal{L}})F-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}$ (4.23)
$\displaystyle\leq{\mathcal{L}}(F\varphi^{-1})-\frac{(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-\frac{\alpha
F^{2}}{\lambda_{1}^{4}}(\nabla_{1}h_{11})^{2}-\sum_{k>\mu}\frac{2\alpha
F^{2}(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})}$ (4.24)
$\displaystyle\quad-F^{2}\left(\frac{f_{11}}{f}-\frac{f_{1}^{2}}{f^{2}}-n\alpha+1\right)-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}.$ (4.25)

Then the conclusion follows from

$\displaystyle\frac{f_{11}}{f}-\frac{f_{1}^{2}}{f^{2}}-n\alpha+1\geq-C\left(n,\alpha,\|f\|_{C^{2}(\mathbb{S}^{n})},\min_{\mathbb{S}^{n}}f\right).$
(4.26)

∎

We are now ready to establish a uniform upper bound for principal curvatures.

###### Proof of Theorem 1.4.

Since $M_{t}$ shrinks under the flow (1.4), we have

$\displaystyle R:=4\sup_{0\leq t\leq
T}\sup_{M_{t}}|x|^{2}=4\sup_{M_{0}}|x|^{2}<\infty.$ (4.27)

We consider a continuous function

$\displaystyle w:=\frac{tfK^{\alpha}\lambda_{\min}^{-1}}{R-|x|^{2}}.$ (4.28)

Since $w(\cdot,0)=0$, we can take its maximum point $(x_{0},t_{0})$ such that
$w\leq w_{0}:=w(x_{0},t_{0})$ holds for $t\leq t_{0}$. Clearly, $t_{0}>0$.

We claim that $w\leq C(1+t)$ holds for some $C$ depending only on $\alpha$,
$n$, $T$, $\|f\|_{C^{2}(\mathbb{S}^{n})}$, $\min_{\mathbb{S}^{n}}f$, and $R$.
If the claim is true, then by using
$K^{\alpha}\lambda_{\min}^{-1}\geq\lambda_{\min}^{(n-1)\alpha-1}\lambda_{\max}^{\alpha}$,
$\alpha\leq 1/(n-1)$, and C, we can obtain the desired result as follow.

$\displaystyle\lambda_{\max}^{\alpha}\leq
C(1+t^{-1})\lambda_{\min}^{1-(n-1)\alpha}\leq
C(1+t^{-1})K^{\frac{1-(n-1)\alpha}{n}}\leq
C(1+t^{-1})^{\frac{2+\alpha}{n\alpha+1}}.$ (4.29)

Note that we can drop the dependence on $T$, because we can obtain $T\leq
C(\alpha,n,\text{diam}M_{0},\min_{\mathbb{S}^{n}}f)$ by applying the
comparison principle with a shrinking ball satisfying $x_{t}=-(\min
f)K^{\alpha}\nu$.

To prove the claim, we define

$\varphi:=\frac{tfK^{\alpha}w_{0}^{-1}}{R-|x|^{2}}.$ (4.30)

Then, observing $w\leq w_{0}$ we have $\varphi\leq\lambda_{\min}$ on $M_{t}$
for $t\in[0,t_{0}]$ as well as
$\varphi(x_{0},t_{0})=\lambda_{\min}(x_{0},t_{0})$. Hence, the smooth function
$\varphi$ satisfies the assumptions in Lemma 4.1. We choose a chart satisfying
$g_{ij}=\delta_{ij}$ and $h_{ij}=\lambda_{i}\delta_{ij}$ at $(x_{0},t_{0})$,
where
$\lambda_{1}=\cdots=\lambda_{\mu}<\lambda_{\mu+1}\leq\cdots\leq\lambda_{n}$.
Then, by Lemma 4.2 we have

$\displaystyle\partial_{t}(F\varphi^{-1})$
$\displaystyle\leq{\mathcal{L}}(F\varphi^{-1})-\frac{\alpha
F^{2}}{\lambda_{1}^{4}}(\nabla_{1}h_{11})^{2}-\sum_{k>\mu}\frac{2\alpha
F^{2}(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})}$ (4.31)
$\displaystyle\quad-\frac{(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+CF^{2}$ (4.32)

at $(x_{0},t_{0})$. Since $F\varphi^{-1}:=w_{0}t^{-1}(R-|x|^{2})$, the
equation of $|x|^{2}$ in Lemma 3.1 implies

$\displaystyle 0$
$\displaystyle\leq\frac{w_{0}}{t_{0}^{2}}(R-|x_{0}|^{2})+\frac{2(n\alpha-1)w_{0}}{t_{0}}Fu-\frac{2\alpha
w_{0}}{t_{0}}F\operatorname{tr}(b)-\frac{2w_{0}}{t_{0}}K^{\alpha}f^{k}\langle
x_{0},\nabla_{k}x\rangle$ (4.33) $\displaystyle\quad-\frac{\alpha
F^{2}}{\lambda_{1}^{4}}(\nabla_{1}h_{11})^{2}-\sum_{k>\mu}\frac{2\alpha
F^{2}(\nabla_{k}h_{11})^{2}}{\lambda_{1}^{3}(\lambda_{k}-\lambda_{1})}-\frac{(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+CF^{2}.$ (4.34)

at $(x_{0},t_{0})$. We define $I_{k}=F\lambda_{1}^{-2}\nabla_{k}h_{11}$ for
$k=1,\dots,n$ so that we obtain

$\displaystyle 0$
$\displaystyle\leq\frac{F}{t_{0}\lambda_{1}}+\frac{2(n\alpha-1)F^{2}u-2\alpha
F^{2}\operatorname{tr}(b)-2FK^{\alpha}f^{k}\langle
x_{0},\nabla_{k}x\rangle}{\lambda_{1}(R-|x_{0}|^{2})}$ (4.35)
$\displaystyle\quad-\alpha
I_{1}^{2}-\sum_{k>\mu}\frac{2\alpha\lambda_{1}I_{k}^{2}}{(\lambda_{k}-\lambda_{1})}-\frac{(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+CF^{2}.$ (4.36)

Then, remembering the definition of $w$, we multiply the above inequality by
$t_{0}^{2}$ to derive

$\displaystyle 0$
$\displaystyle\leq(R^{2}-|x|^{2})w_{0}+2t_{0}(n\alpha-1)Fuw_{0}-2\alpha
t_{0}F\operatorname{tr}(b)w_{0}-2t_{0}K^{\alpha}f^{k}\langle
x_{0},\nabla_{k}x\rangle w_{0}$ (4.37) $\displaystyle\quad-\alpha
t_{0}^{2}I_{1}^{2}-\sum_{k>\mu}\frac{2\alpha
t_{0}^{2}\lambda_{1}I_{k}^{2}}{\lambda_{k}-\lambda_{1}}-\frac{t_{0}^{2}(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2t_{0}^{2}\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+Ct_{0}^{2}F^{2}.$ (4.38)

Note that we have $|x|\leq R$ and $u\leq R^{\frac{1}{2}}$. Also, C says
$tF\leq C$. Hence,

$\displaystyle 0$ $\displaystyle\leq-2\alpha
t_{0}F\operatorname{tr}(b)w_{0}-\alpha
t_{0}^{2}I_{1}^{2}-\sum_{k>\mu}\frac{2\alpha
t_{0}^{2}\lambda_{1}}{\lambda_{k}-\lambda_{1}}I_{k}^{2}$ (4.39)
$\displaystyle\quad-\frac{t_{0}^{2}(\nabla_{1}F)^{2}}{\lambda_{1}^{2}}-2t_{0}^{2}\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}+Cw_{0}+C.$ (4.40)

To replace $\nabla\varphi^{-1}$ and $\nabla F$, remembering
$\nabla_{k}h_{11}=F^{-1}\lambda_{1}^{2}I_{k}$, we observe

$\displaystyle\nabla\varphi^{-1}=-\lambda_{1}^{-2}\nabla
h_{11}=-F^{-1}I_{k}\nabla_{k}x,$
$\displaystyle\nabla(t_{0}F\varphi^{-1})=-w_{0}\nabla|x|^{2}.$ (4.41)

Therefore, we have

$t_{0}\lambda_{1}^{-1}\nabla F=t_{0}I_{k}\nabla_{k}x-w_{0}\nabla|x|^{2}.$
(4.42)

This implies

$-t_{0}^{2}\lambda_{1}^{-2}(\nabla_{1}F)^{2}=-t_{0}^{2}I_{1}^{2}+2t_{0}w_{0}(\nabla_{1}|x|^{2})I_{1}-w_{0}^{2}|\nabla_{1}|x|^{2}|^{2},$
(4.43)

and

$\displaystyle-2t_{0}^{2}\left\langle{\nabla
F},{\nabla\varphi^{-1}}\right\rangle_{W}$
$\displaystyle=2t_{0}\left\langle{t_{0}\lambda_{1}I_{k}\nabla_{k}x-\lambda_{1}w_{0}\nabla|x|^{2}},{F^{-1}I_{k}\nabla_{k}x}\right\rangle_{W}$
(4.44) $\displaystyle=\sum_{k=1}^{n}\frac{2\alpha
t_{0}^{2}\lambda_{1}}{\lambda_{k}}I_{k}^{2}-\sum_{k=1}^{n}\frac{2\alpha
t_{0}\lambda_{1}\nabla_{k}|x|^{2}}{\lambda_{k}}w_{0}I_{k}.$ (4.45)

Since Lemma 4.1 implies $I_{k}=0$ for $1<k\leq\mu$, we have

$\displaystyle 0$ $\displaystyle\leq-2\alpha
t_{0}F\operatorname{tr}(b)w_{0}+(\alpha-1)t_{0}^{2}I_{1}^{2}-\sum_{k>\mu}\left(\frac{2\alpha
t_{0}^{2}\lambda_{1}^{2}}{\lambda_{k}(\lambda_{k}-\lambda_{1})}I_{k}^{2}+\frac{2\alpha
t_{0}\lambda_{1}\nabla_{k}|x|^{2}}{\lambda_{k}}w_{0}I_{k}\right)$ (4.46)
$\displaystyle\quad+2(1-\alpha)t_{0}w_{0}(\nabla_{1}|x|^{2})I_{1}-w_{0}^{2}|\nabla_{1}|x|^{2}|^{2}+Cw_{0}+C.$
(4.47)

Since $0<\alpha\leq\frac{1}{n-1}\leq 1$, we have

$\displaystyle(\alpha-1)t_{0}^{2}I_{1}^{2}+2(1-\alpha)t_{0}w_{0}\nabla_{1}|x|^{2}I_{1}\leq(1-\alpha)w_{0}^{2}|\nabla_{1}|x|^{2}|^{2}.$
(4.48)

Also, it follows from $\lambda_{k}-\lambda_{1}\leq\lambda_{k}$ that

$\displaystyle-\sum_{k>\mu}\left(\frac{2\alpha
t_{0}^{2}\lambda_{1}^{2}}{\lambda_{k}(\lambda_{k}-\lambda_{1})}I_{k}^{2}+\frac{2\alpha
t_{0}\lambda_{1}\nabla_{k}|x|^{2}}{\lambda_{k}}w_{0}I_{k}\right)\leq\sum_{k>\mu}\frac{\alpha}{2}w_{0}^{2}|\nabla_{k}|x|^{2}|^{2}.$
(4.49)

Furthermore, by using
$\operatorname{tr}(b)=\sum_{k=1}^{n}\lambda_{k}^{-1}\geq\lambda_{1}^{-1}$, we
have

$\displaystyle-2\alpha
t_{0}F\operatorname{tr}(b)w_{0}\leq-2\alpha(R-|x|^{2})w_{0}^{2}.$ (4.50)

Putting these all together, (4.46) becomes

$\displaystyle 0$
$\displaystyle\leq\Big{(}-2\alpha(R-|x|^{2})+\sum_{k>\mu}\frac{\alpha}{2}|\nabla_{k}|x|^{2}|^{2}-\alpha|\nabla_{1}|x|^{2}|^{2}\Big{)}w_{0}^{2}+Cw_{0}+C.$
(4.51)

Since
$|x|^{2}-u^{2}=\sum_{k=1}^{n}\left\langle{x},{\nabla_{k}x}\right\rangle^{2}$,
the coefficient of $w_{0}^{2}$ satisfies

$\displaystyle-2\alpha(R-|x|^{2})+\sum_{k>\mu}\frac{\alpha}{2}|\nabla_{k}|x|^{2}|^{2}-\alpha|\nabla_{1}|x|^{2}|^{2}\leq-2\alpha(R-|x|^{2})+2\alpha|x|^{2}\leq-\alpha
R$ (4.52)

by our choice of $R=4\sup_{0\leq t\leq T}\sup_{M_{t}}|x|^{2}$. Therefore, we
arrive at

$\displaystyle 0\leq-\alpha Rw_{0}^{2}+Cw_{0}+C.$ (4.53)

This completes the proof. ∎

##  5\. Proof of Theorem 1.1

We first extend the result of Theorem 1.4 to viscosity solutions using an
approximation argument (see Definition 2.1 for the definition of viscosity
solutions).

######  Theorem 5.1.

Let $\alpha\in(0,\frac{1}{n-1}]$, and let $M_{0}=\partial\Omega_{0}$ be the
boundary of a convex body $\Omega_{0}$ in $\mathbb{R}^{n+1}$. Suppose $f$ is a
positive smooth function on $\mathbb{S}^{n}$. Then, there exists a unique
viscosity solution $\\{\Omega_{t}\\}_{0<t\leq T^{\prime}}$ to the flow (1.4)
which converges to $\Omega_{0}$ in Hausdorff distance as $t\to 0$. For each
$t\in(0,T^{\prime}]$, the corresponding hypersurface
$M_{t}=\partial\Omega_{t}$ is of class $C^{1,1}$. Moreover, there exist
constants $C,T$ depending only on $n$, $\alpha$, $\operatorname{diam}M_{0}$,
$\rho_{-}(M_{0})$, $\|f\|_{C^{2}(\mathbb{S}^{n})}$, and
$\min_{\mathbb{S}^{n}}f$ such that

$\displaystyle\operatorname{ess\,sup}_{M_{t}}\lambda_{i}\leq
C(1+t^{-\frac{1+\alpha}{n\alpha^{2}}}),\quad\text{for}\quad 0<t\leq T,\quad
i=1,\ldots,n.$ (5.1)

###### Proof.

By B, it suffices to prove (5.1). Consider an increasing family of smooth,
closed, strictly convex hypersurfaces $M^{\varepsilon}$ that converge to
$M_{0}$ in Hausdorff distance as $\varepsilon\rightarrow 0$. By A, there
exists a unique solution $M^{\varepsilon}_{t}$ to (1.4) with initial data
$M^{\varepsilon}$.

We choose an origin for ${\mathbb{R}}^{n+1}$ and radii $0<r<R$ such that, for
sufficiently small $\varepsilon$, the hypersurfaces $M^{\varepsilon}$ contain
the ball $B_{r}(0)$ and are enclosed by the ball $B_{R}(0)$. Since
$M^{\varepsilon}_{t}$ shrinks, every $M^{\varepsilon}_{t}$ with $t\geq 0$ is
enclosed by $B_{R}(0)$. Also, there is a small time $T>0$ such that all
hypersurfaces $M_{t}^{\varepsilon}$ for $t\in[0,T^{\prime}]$ enclose the ball
$B_{r/2}(0)$, by the comparison principle with a shrinking ball
$B_{\rho(t)}(0)$ where
$\rho^{\prime}=-(\max_{\mathbb{S}^{n}}f)\rho^{-n\alpha}$.

We recall the identity $|x|^{2}=u^{2}+|\bar{\nabla}_{\mathbb{S}^{n}}u|^{2}$ of
convex hypersurface, where $x$ is the position vector, $u(\nu)$ is the support
function, and $\bar{\nabla}_{\mathbb{S}^{n}}$ is the Levi-Civita connection of
the unit sphere metric. Since the hypersurfaces $M^{\varepsilon}$ are convex
and have bounded diameter, the identity implies that the gradients of their
support functions $u^{\varepsilon}(z,t)$ are uniformly Lipschitz on
$\mathbb{S}^{n}$. Moreover, according to C, the speeds are bounded over
$[\delta,T]$ for any $\delta>0$. Thus, the functions $u^{\varepsilon}$ are
uniformly Hölder continuous on the space-time
$\mathbb{S}^{n}\times[\delta,T]$. By the Arzela–Ascoli theorem, we can extract
a subsequence $\varepsilon_{k}$ such that $u^{\varepsilon_{k}}$ converges
uniformly to a limit function
$u:\mathbb{S}^{n}\times(0,T]\rightarrow{\mathbb{R}}$ with the same estimates.
By the Blaschke selection theorem, each function $u(\cdot,t)$ is the support
function of a convex hypersurface $M_{t}$. Applying Theorem 1.4 to
$M^{\varepsilon}_{t}$, we obtain the desired estimate (5.1) for the
hypersurfaces $M_{t}$. Finally, it can be shown by the comparison principle
that the hypersurfaces $M_{t}$ is a viscosity solution to (1.4). ∎

######  Lemma 5.2.

If $p\in(-\infty,-n+2]$ and $f$ is a positive smooth function on
$\mathbb{S}^{n}$, then any generalized solution $\Sigma$ to (1.2) is a self-
similar solution to the flow (1.4) with $\alpha=\frac{1}{1-p}$ and
$f^{\alpha}$ instead of $f$.

###### Proof.

Consider the function $a(t)=(1-(n\alpha+1)t)^{\frac{1}{n\alpha+1}}$, and let
$M_{0}=\Sigma$ be a generalized solution to the $L_{p}$ Minkowski problem with
$p\leq-n+2$. We define $M_{t}=a(t)M_{0}$. Since the viscosity solution to
(1.4) is unique by Theorem 5.1, we only need to prove that $M_{t}$ is a
viscosity solution to the flow (1.4).

Let $M_{0}^{-}$ be a strictly convex, closed, smooth hypersurface which is
enclosed by $M_{0}$. We may assume $M_{0}^{-}\cap M_{0}=\emptyset$. Otherwise,
we consider $(1-\varepsilon)M_{0}^{-}$ and then take $\varepsilon\rightarrow
0$. Assume that the evolution $M_{t}^{-}$ of $M_{0}^{-}$ under the flow (1.4)
touches $M_{t}$ for the first time $t=t_{0}\in(0,1/(n\alpha+1))$ at $x_{0}\in
M_{t_{0}}^{-}\cap M_{t_{0}}$. Let $\nu_{0}$ be the normal of $M_{t_{0}}^{-}$
at $x_{0}$. If $u^{-}(\cdot,t)$ and $u(\cdot,t)$ are the support functions of
$M_{t}^{-}$ and $M_{t}$, respectively, then
$u^{-}(\nu_{0},t_{0})=u(\nu_{0},t_{0})\geq 0$ since the origin is enclosed by
$M_{t}$ for all $0\leq t\leq 1/(n\alpha+1)$. If
$u^{-}(\nu_{0},t_{0})=u(\nu_{0},t_{0})>0$, then $u>u^{-}>0$ in a neighborhood
of $\nu_{0}$ and $0\leq t<t_{0}$. Then by the standard regularity theory of
Caffarelli [16], $u$ and $u^{-}$ satisfy (1.2) locally in a classical sense,
which contradicts to $u^{-}(\nu_{0},t_{0})=u(\nu_{0},t_{0})$ by the comparison
principle.

If $u^{-}(\nu_{0},t_{0})=u(\nu_{0},t_{0})=0$, then $x_{0}\in M_{t}$ for all
$t\in[0,t_{0}]$ by definition of $M_{t}$. However, $M_{t_{0}}^{-}$ is strictly
enclosed by $M_{t}^{-}$ for any $t_{0}>t\geq 0$, so $x_{0}$ lies in the
interior of $M_{t}^{-}$ for any $t\in[0,t_{0})$, implying $M_{0}\cap
M_{0}^{-}\not=\emptyset$, which is a contradiction. This proves that $M_{t}$
satisfies the first condition in Definition 2.1.

To prove that $M_{t}$ satisfies the second condition in Definition 2.1, let
$M_{0}^{+}$ be a strictly convex, closed, smooth hypersurface which encloses
$M_{0}$. As before, we may assume $M_{0}^{+}\cap M_{0}=\emptyset$. Otherwise,
we consider $(1+\varepsilon)M_{0}^{+}$ and then take $\varepsilon\rightarrow
0$.

Suppose that the evolution $M_{t}^{+}$ of $M_{0}^{+}$ under the flow (1.4)
touches $M_{t}$ for the first time $t=t_{0}\in(0,1/(n\alpha+1))$ at $x_{0}\in
M_{t_{0}}^{+}\cap M_{t_{0}}$. Let $\nu_{0}$ be the normal of $M_{t_{0}}^{+}$
at $x_{0}$. If $u^{+}(\cdot,t)$ is the support function of $M_{t}^{+}$, then
$u^{+}(\nu_{0},t_{0})=u(\nu_{0},t_{0})\geq 0$. If
$u^{+}(\nu_{0},t_{0})=u(\nu_{0},t_{0})>0$, then the standard regularity theory
with the comparison principle gives a contradiction as above.

Suppose that $u^{+}(\nu_{0},t_{0})=u(\nu_{0},t_{0})=0$. If $x_{0}$ is not the
origin, then the line segment $l$ connecting $x_{0}$ and the origin is
contained in $M_{t_{0}}$. Consequently, $l$ is enclosed by $M_{t_{0}}^{+}$.
However, as $M_{t_{0}}^{+}$ is smooth, the normal vector $\nu_{0}$ must be
orthogonal to $l$. Thus, $l$ is also contained in $M_{t_{0}}^{+}$, which
contradicts the strict convexity of $M_{t_{0}}^{+}$. Therefore, $x_{0}$ must
be the origin.

We can rotate the hypersurface $M_{t_{0}}$ if necessary, so that
$\nu_{0}=(0,\ldots,0,-1)$. This implies that $M_{t_{0}}$ lies in the half-
space $\\{x\in{\mathbb{R}}^{n+1}:x_{n+1}\geq 0\\}$. Next, we consider a strip
$S_{\varepsilon}=\\{(x^{\prime},x_{n+1})\in{\mathbb{R}}^{n+1}:0\leq
x_{n+1}\leq\varepsilon\\}$ for each small $\varepsilon>0$. Within the strip
$S_{\varepsilon}$, both hypersurfaces $M_{t_{0}}$ and $M_{t_{0}}^{+}$ are
represented by graphs of functions $U$ and $U^{+}$ respectively over subsets
of ${\mathbb{R}}^{n}\times\\{0\\}$. Since $M_{t_{0}}^{+}$ encloses
$M_{t_{0}}$, we have $0\leq U^{+}\leq U$, and the equality only holds at
$x^{\prime}=0$. As $M_{t_{0}}^{+}$ is a strictly convex and smooth
hypersurface, we can find a constant $k$ such that
$U^{+}\geq\frac{k}{2}|x^{\prime}|^{2}$.

Let $G$ be the image of normals $M_{t_{0}}\cap S_{\varepsilon}$. By convexity,
$G$ contains the ball of radius $\sqrt{2k\varepsilon}$ since
$U\geq\frac{k}{2}|x^{\prime}|^{2}$. Then we have

$\displaystyle\int_{G}f\geq C_{n}^{-1}f_{\min}(k\varepsilon)^{n/2}$ (5.2)

for some dimensional constant $C_{n}>0$.

Observe that $p\leq 2-n\leq 0$ and that $u\leq\sqrt{2\varepsilon k^{-1}}$ on
$G$. Then

$\displaystyle\int_{G}u^{1-p}\,\mathrm{d}S({M_{t_{0}}},\cdot)\leq(2\varepsilon
k^{-1})^{-\frac{p}{2}}\int_{G}u\,\mathrm{d}S({M_{t_{0}}},\cdot)=(n+1)(2\varepsilon
k^{-1})^{-\frac{p}{2}}V(M_{t_{0}}\cap S).$ (5.3)

Since $V(M_{t_{0}}\cap S)\leq C_{n}\varepsilon(2\varepsilon
k^{-1})^{\frac{n}{2}}$, we obtain

$\displaystyle\int_{G}u^{1-p}\,\mathrm{d}S({M_{t_{0}}},\cdot)\leq
C(n,k,p)\varepsilon^{\frac{n-p+2}{2}}.$ (5.4)

This, together with (5.2) and the fact that $M_{t}$ is a generalized solution
to (1.2), implies

$\displaystyle\varepsilon^{\frac{2-p}{2}}\geq C(n,k,p)f_{\min},$ (5.5)

which is a contradiction if we take $\varepsilon\rightarrow 0$. This proves
that $M_{t}$ satisfies the second condition in Definition 2.1. ∎

We next show that the inradius of a generalized solution (1.2) is controlled
by its diameter.

######  Lemma 5.3.

For $p\leq-n+2$, if a convex body $\Omega$ is a generalized solution to (1.2),
then there is a constant dependent only on $n$, $p$, and $diam(\Omega)$ such
that $\rho_{-}(\Omega)\geq c$.

###### Proof.

Let $E$ be the John ellipsoid associated to $\Omega$ so that
$E\subset\Omega\subset(n+1)E$. This is the ellipsoid contained in $\Omega$
with maximal volume. If the principal radii of $E$ are denoted by
$0<r_{1}\leq\cdots\leq r_{n+1}$, then
$r_{1}\leq\rho_{-}(E)\leq\rho_{-}(\Omega)$. With the use of $r_{n+1}\leq
u_{\max}$, it follows that $V(E)\leq 2^{n+1}r_{1}\cdots r_{n+1}\leq
2^{n+1}r_{1}u_{\max}^{n}$. Since $V(E)\geq C_{n}V(\Omega)$, we can infer that

$\displaystyle\rho_{-}(\Omega)\geq r_{1}\geq C_{n}V(\Omega)u_{\max}^{-n}.$
(5.6)

To complete the proof, we aim to show that the volume of $\Omega$ has a
positive lower bound dependent only on the diameter of $\Omega$. Note that
$p\leq-n+2\leq 0$. Since $\Omega$ is a generalized solution to (1.2), we can
see that

$\displaystyle\int_{\mathbb{S}^{n}}f=\int_{\mathbb{S}^{n}}u^{1-p}\,\mathrm{d}S(\Omega,\cdot)\leq
u_{\max}^{-p}\int_{\mathbb{S}^{n}}u\,\mathrm{d}S(\Omega,\cdot)=(n+1)u_{\max}^{-p}V(\Omega)$
(5.7)

which gives a positive lower bound on the volume of $\Omega$ in terms of
diameter. Combining this with (5.6), we obtain the desired result. ∎

Finally we prove the main result of the paper.

###### Proof of Theorem 1.1.

As $\Sigma$ is a generalized solution to (1.2), it follows from Lemma 5.2 that
$\Sigma$ is a self-similar solution to the flow (1.4) with
$\alpha=\frac{1}{1-p}$ and $f^{\alpha}$ instead of $f$. Specifically,
$M_{t}=a(t)\Sigma$ where $a(t)=(1-(n\alpha+1)t)^{\frac{1}{n\alpha+1}}$ is the
solution to the flow (1.4) starting from $\Sigma$ with maximal existence time
$T_{\max}=1/(n\alpha+1)$. By using Lemma 5.3, we have that $\rho_{-}(\Sigma)$
is controlled in terms of $n$, $\alpha$, and $\operatorname{diam}(\Sigma)$.
Therefore, the dependence $\rho_{-}(M_{0})$ on the constant $C$ can be
absorbed. Hence, applying Theorem 5.1 with some small $T<T_{\max}$ only
depending on $\alpha,n,\max_{\mathbb{S}^{n}}f,\operatorname{diam}(\Sigma)$, we
obtain

$\displaystyle\lambda_{i}(\Sigma)=a(T)\lambda_{i}(M_{T})\leq a(T)C(1+T^{-1}),$
(5.8)

where the constant
$C(\alpha,n,\|f\|_{C^{2}(\mathbb{S}^{n})},\min_{\mathbb{S}^{n}}f,\operatorname{diam}(\Sigma))$
is given in Theorem 5.1. This completes the proof. ∎

##  6\. Proof of Theorem 1.2

We finally prove Theorem 1.2 in this section. The proofs of Theorem 1.2 (i)
and (ii) are provided in Section 6.1 and Section 6.2, respectively.

###  6.1. Contraction mapping

In this section, we provide the proof of Theorem 1.2 (i). Indeed, we prove a
slightly more general statement:

######  Theorem 6.1.

If $-n+1<p<1$, then there exists a generalized solution $\Sigma$ to (1.2) such
that $\Sigma$ is a hypersurface of $($at most $)$ class $C^{k,\gamma}$ with
$\gamma\in(0,1]$ for $k+\gamma=\frac{n+p}{n+p-1}$, and $f$ is a positive
smooth function.

Suppose that a generalized solution to the $L_{p}$ Minkowski problem includes
the graph of a convex function $v:U(\subset\mathbb{R}^{n})\to\mathbb{R}$.
Suppose $v\in C^{2}(V)$ for some $V\subset U$. Then,

$\tfrac{\det
D^{2}v}{(1+|Dv|^{2})^{\frac{n+2}{2}}}=f^{-1}(\nu)\Big{(}\tfrac{x\cdot
Dv-v}{\sqrt{1+|Dv|^{2}}}\Big{)}^{1-p}$ (6.1)

holds in $V$, where $\nu=\frac{(Dv,-1)}{\sqrt{1+|Dv|^{2}}}$. We assume that
$f=1$ on the graph $\\{(x,v(x)):x\in U\\}$, and $v:B_{1+R}(0)\setminus
B_{1}(0)\to\mathbb{R}$ is a radial function. Then, the one-variable function
$\bar{v}(r)$, defined by

$v(x)=\bar{v}(|x|-1),$ (6.2)

satisfies

$\bar{v}_{rr}\bar{v}_{r}^{n-1}=(\bar{v}_{r}+r\bar{v}_{r}-\bar{v})^{1-p}(1+r)^{n-1}(1+\bar{v}_{r}^{2})^{\frac{n+p+1}{2}}$
(6.3)

for $r\in(0,R)$. Note that if $|r|\ll 1$ and $|\bar{v}|\ll|\bar{v}_{r}|$ then
$\bar{v}_{rr}\bar{v}_{r}^{n-1}\approx\bar{v}_{r}^{1-p}$. Hence, it is useful
to consider the solution

$h(r):=\tfrac{m}{1+m}m^{\frac{1}{m}}r^{\frac{1+m}{m}},\qquad\text{where}\;\;m:=n+p-1,$
(6.4)

to the model equation

$h_{rr}=h_{r}^{1-m}.$ (6.5)

In this section, we will show that given $p,n$ with $m>0$, there are $r_{0}>0$
and $\bar{v}:I_{0}\to\mathbb{R}$, where $I_{0}:=(0,r_{0}]$, such that
$\bar{v}(0)=\bar{v}_{r}(0)=0$, $\bar{v}\in C^{2}(I_{0})$, and

$\displaystyle\bar{v}=(1+O(r^{\delta}))h,$
$\displaystyle\bar{v}_{r}=(1+O(r^{\delta}))h_{r},$
$\displaystyle\bar{v}_{rr}=(1+O(r^{\delta}))h_{rr},$ (6.6)

where $\delta:=\min\\{1,\frac{2}{m}\\}$. This will lead us to the proof of
Theorem 1.2 (i).

Given $\varphi\in C^{k}{(I)}$ with an interval $I$, we define a weighted norm
on $C^{k}_{w}(I)$ by

$\|\varphi\|_{C^{k}_{w}(I)}=\max_{l=0,\cdots,k}\|r^{l-1}h_{r}^{-1}\varphi^{(l)}\|_{L^{\infty}(I)},$
(6.7)

where $\varphi^{(l)}$ is the $l$-th order derivative of $\varphi$. In
addition, we define some linear operators

$\displaystyle[\varphi]_{r}:=\frac{\varphi_{r}}{h_{r}},$
$\displaystyle[\varphi]_{s}:=[\varphi]_{r}-\frac{\varphi}{rh_{r}},$
$\displaystyle[\varphi]_{rr}:=\frac{\varphi_{rr}}{h_{rr}}.$ (6.8)

Let $w=\bar{v}-h$, and manipulate
$\bar{v}_{rr}\bar{v}_{r}^{n-1}-h_{rr}h_{r}^{n-1}$ by using (6.3) and (6.5) to
derive

$\displaystyle
h_{r}^{n-1}w_{rr}+(n-1)h_{r}^{n-2}h_{rr}w_{r}=(1-p)h_{r}^{-p}w_{r}+h_{r}^{1-p}E[w],$
(6.9)

where

$\displaystyle E[w]:=P_{1}[w]+P_{2}[w]+Q[w]+R_{1}[w]-R_{2}[w]-R_{3}[w],$
(6.10)

with

$\displaystyle
P_{1}[w]:=((1+r)^{n-1}-1)(1+h_{r}^{2}(1+[w]_{r})^{2})^{\frac{m+2}{2}}(1+\tfrac{1}{m+1}r+[w]_{r}+r[w]_{s})^{1-p},$
(6.11) $\displaystyle
P_{2}[w]:=((1+h_{r}^{2}(1+[w]_{r})^{2})^{\frac{m+2}{2}}-1)(1+\tfrac{1}{m+1}r+[w]_{r}+r[w]_{s})^{1-p},$
(6.12) $\displaystyle
Q[w]:=(1+\tfrac{1}{m+1}r+[w]_{r}+r[w]_{s})^{1-p}-(1+[w]_{r})^{1-p},$ (6.13)

and

$\displaystyle R_{1}[w]:=(1+[w]_{r})^{1-p}-1-(1-p)[w]_{r},$ (6.14)
$\displaystyle R_{2}[w]:=[w]_{rr}((1+[w]_{r})^{n-1}-1),$ (6.15) $\displaystyle
R_{3}[w]:=(1+[w]_{r})^{n-1}-1-(n-1)[w]_{r}.$ (6.16)

Note that we utilized (6.5) for formulating $R_{2}$ and $R_{3}$. Also,
$Q\equiv R_{1}\equiv 0$ holds in the case $p=1$. Next, by using (6.5) and
$h_{r}=(mr)^{\frac{1}{m}}$, we can simplify the equation of $w$ as

$Lw:=(r^{1-\frac{1}{m}}w_{r})_{r}=m^{\frac{1}{m}-1}E[w].$ (6.17)

Let us provide the main result in this section.

######  Theorem 6.2.

Given $p,n$ with $m>0$, there are $C_{0},r_{0}>0$ and $w\in C^{2}(I_{0})$ with
$I_{0}:=(0,r_{0}]$ such that $w$ is a solution to (6.17) on $I_{0}$ satisfying

$\|w\|_{C^{2}_{w}((0,r])}\leq
C_{0}r^{\delta}\leq\tfrac{1}{10}\min\\{m,m^{-1}\\}$ (6.18)

for every $r\leq r_{0}$, where $\delta:=\min\\{1,\frac{2}{m}\\}$.

By using this theorem, we can prove Theorem 1.2 (i), which follows from
Theorem 6.1.

###### Proof of Theorem 6.1.

We recall $w$ in Theorem 6.2 so that we define $\bar{v}=h+w$ and then define
$v:B_{1+r_{0}}(0)\to\mathbb{R}$ by

  1. (1)

$v(x)\equiv 0$ for $|x|\leq 1$,

  2. (2)

$v(x)=\bar{v}(|x|-1)$ for $|x|\in(1,r_{0}+1)$.

Then, (6.18) implies
$\frac{|w|}{h},\frac{|w_{r}|}{h_{r}},\frac{|w_{rr}|}{h_{rr}}\leq\frac{1}{2}$,
and therefore $v$ is a positive and strictly convex function on
$B_{1+r_{0}}(0)\setminus B_{1}(0)$. Also, $v(x)$ is smooth in
$\\{1<|x|<1+r_{0}\\}$ by the standard regularity theorems for the Monge–Ampère
type equations. Thus, $v$ is of class $C^{k,\gamma}$ with $\gamma\in(0,1]$ for
$k+\gamma=1+\frac{1}{m}$. Notice that the assumption $p<1$ has not been used
so far.

We next choose a rotationally symmetric convex body
$\Omega\subset\mathbb{R}^{n}\times[0,+\infty)$ such that

  1. (1)

$\partial\Omega\cap\\{x_{n+1}\leq\frac{1}{2}\bar{v}(r_{0})\\}=\\{(x,v(x)):v(x)\leq\frac{1}{2}\bar{v}(r_{0})\\}$,

  2. (2)

$\partial\Omega$ is smooth, and its Gauss curvature is positive in
$\\{x_{n+1}>0\\}$.

To prove that $\Omega$ is a generalized solution of (1.2), we need to show
(1.3). Note that we have

$\int_{E\cap\\{v>0\\}}u^{1-p}\,\mathrm{d}S(\Omega,z)=\int_{E\cap\\{v>0\\}}f\,\mathrm{d}\sigma\quad\text{for
all Borel sets }E\subset\mathbb{S}^{n}$ (6.19)

for some positive function $f\in C^{\infty}(\mathbb{S}^{n})$, where $u$ is the
support function of $\Omega$. Since $u=0$ on $\\{v=0\\}$ and $p<1$, we obtain

$\int_{E\cap\\{v=0\\}}u^{1-p}\,\mathrm{d}S(\Omega,z)=0.$ (6.20)

Moreover, since $\\{v=0\\}\subset\\{-e_{n}\\}$ we have

$\int_{E\cap\\{v=0\\}}f\,\mathrm{d}\sigma=0.$ (6.21)

Therefore, it follows from (6.20) and (6.21) that

$\int_{E}u^{1-p}\,\mathrm{d}S(\Omega,z)=\int_{E}f\,\mathrm{d}\sigma\quad\text{for
all Borel sets }E\subset\mathbb{S}^{n},$ (6.22)

which proves (1.2). We finish the proof by observing that $\partial\Omega$ is
a hypersurface of at most class $C^{k,\gamma}$ with $\gamma\in(0,1]$ for
$k+\gamma=1+\frac{1}{m}$ as the regularity of $v$. ∎

To prove Theorem 6.2, we employ the contraction mapping. We define
$w_{0}\equiv 0$ and

$w_{1}(r):=\int_{0}^{r}s^{\frac{1}{m}-1}\int_{0}^{s}m^{\frac{1}{m}-1}E(w_{0})(t)\,\mathrm{d}t\,\mathrm{d}s,$
(6.23)

which solves $Lw_{1}=m^{\frac{1}{m}-1}E(w_{0})$. Then, for $i\in\mathbb{N}$,
we inductively define

$w_{i+1}(r):=w_{i}(r)+\int_{0}^{r}s^{\frac{1}{m}-1}\int_{0}^{s}m^{\frac{1}{m}-1}(E(w_{i})(t)-E(w_{i-1})(t))\,\mathrm{d}t\,\mathrm{d}s$
(6.24)

so that $Lw_{i+1}=m^{\frac{1}{m}-1}E(w_{i})$. Thus, it is important to
estimate the error term $E$.

######  Lemma 6.3.

There is some constant $C_{1}$ depending only on $n,p$ with $m>0$ such that

$|E(w_{0})(r)|\leq C_{1}r^{\delta}$ (6.25)

holds for every $r\leq 1$, where $\delta=\min\\{1,\frac{2}{m}\\}$.

###### Proof.

Observe $R_{1}[0]=R_{2}[0]=R_{3}[0]=0$, $Q[0]=(1+\frac{1}{m+1}r)^{1-p}-1$, and

$\displaystyle
P_{1}[0]=((1+r)^{n-1}-1)(1+h_{r}^{2})^{\frac{m+2}{2}}(1+\tfrac{1}{m+1}r)^{1-p},$
(6.26) $\displaystyle
P_{2}[0]=((1+h_{r}^{2})^{\frac{m+2}{2}}-1)(1+\tfrac{1}{m+1}r)^{1-p}.$ (6.27)

Since $w_{0}:=0$, we can obtain the desired result by the Taylor remainder
theorem. ∎

######  Proposition 6.4.

Given $a,b>0$ and $q\in\mathbb{R}$, the following holds

$|a^{q}-b^{q}|\leq|q||a-b|(a^{q-1}+b^{q-1}).$ (6.28)

###### Proof.

Let $a>b$ and $q>1$. Then, the convexity of $f(x)=x^{q}$ implies

$\tfrac{f(a)-f(b)}{a-b}\leq f^{\prime}(a),$ (6.29)

which is the desired result. In the same manner, we can complete the proof by
using the convexity of $f(x)=x^{q}$ for $q<0$ and the concavity of
$f(x)=x^{q}$ for $0<q<1$. The cases $q=0$ and $q=1$ are trivial. ∎

######  Lemma 6.5.

There is some constant $C_{2}$ depending only on $n,p$ with $m>0$ such that if
some $\varphi,\psi\in C^{2}(I)$, where $I=(0,r]$ for some $r\leq 1$, satisfy
$\|\varphi\|_{C^{2}_{w}(I)},\|\psi\|_{C^{2}_{w}(I)}\leq\frac{1}{4}$, then the
following hold: for $j=1,2$

$\displaystyle|P_{j}[\varphi](r)-P_{j}[\psi](r)|\leq
C_{2}r^{\delta}\|\varphi-\psi\|_{C^{2}_{w}(I)},$ (6.30)

and

$\displaystyle|Q[\varphi](r)-Q[\psi](r)|\leq
C_{2}(r^{\delta}+\|\varphi\|_{C^{2}_{w}(I)}+\|\psi\|_{C^{2}_{w}(I)})\|\varphi-\psi\|_{C^{1}_{w}(I)},$
(6.31)

and for $j=1,2,3$

$\displaystyle|R_{j}[\varphi](r)-R_{j}[\psi](r)|\leq
C_{2}(\|\varphi\|_{C^{2}_{w}(I)}+\|\psi\|_{C^{2}_{w}(I)})\|\varphi-\psi\|_{C^{2}_{w}(I)},$
(6.32)

where $\delta=\min\\{1,\frac{2}{m}\\}$.

###### Proof.

For the purpose of brevity, we define

$\displaystyle X_{w}=1+h_{r}^{2}(1+[w]_{r})^{2},$ $\displaystyle
Y_{w}=1+\tfrac{1}{m+1}r+[w]_{r}+r[w]_{s},$ $\displaystyle Z_{w}=1+[w]_{r}.$
(6.33)

Note that if $\|w\|_{C^{2}_{w}(I)}\leq\frac{1}{4}$ we have

$Y_{w},Z_{w}\geq\tfrac{3}{4}.$ (6.34)

We first show (6.30) for $j=1$. Since $(1+r)^{n-1}\leq 1+Cr$ for some $C(n)$
in $\\{0<r\leq 1\\}$, remembering $1\leq X\leq C$ and $\frac{3}{4}\leq Y\leq
C$ we apply Proposition 6.4 so that

$\displaystyle|P_{1}[\varphi]-P_{1}[\psi]|$ $\displaystyle\leq
Cr\left(\Big{|}X_{\varphi}^{\frac{m+2}{2}}-X_{\psi}^{\frac{m+2}{2}}\Big{|}Y_{\varphi}^{1-p}+X_{\psi}^{\frac{m+2}{2}}\Big{|}Y_{\varphi}^{1-p}-Y_{\psi}^{1-p}\Big{|}\right)$
(6.35) $\displaystyle\leq
Cr\left(|X_{\varphi}-X_{\psi}|+|Y_{\varphi}-Y_{\psi}|\right)$ (6.36)

for some $C(n,p)$. Thus, by observing

$\begin{split}&|X_{\varphi}-X_{\psi}|\leq Ch_{r}^{2}|[\varphi-\psi]_{r}|\leq
Cr^{\frac{2}{m}}\|\varphi-\psi\|_{C^{1}_{w}(I)},\\\
&|Y_{\varphi}-Y_{\psi}|\leq C|[\varphi-\psi]_{r}+r[\varphi-\psi]_{s}|\leq
C\|\varphi-\psi\|_{C^{1}_{w}(I)},\end{split}$ (6.37)

we can obtain (6.30) for $j=1$. Similarly, we can derive

$\displaystyle|P_{2}[\varphi]-P_{2}[\psi]|$
$\displaystyle\leq\Big{|}X_{\varphi}^{\frac{m+2}{2}}-X_{\psi}^{\frac{m+2}{2}}\Big{|}Y_{\varphi}^{1-p}+(X_{\psi}^{\frac{m+2}{2}}-1)\Big{|}Y_{\varphi}^{1-p}-Y_{\psi}^{1-p}\Big{|}$
(6.38) $\displaystyle\leq
C|X_{\varphi}-X_{\psi}|+C(X_{\psi}-1)|Y_{\varphi}-Y_{\psi}|.$ (6.39)

Thus, combining with $X_{\psi}-1=h_{r}^{2}Z_{\psi}^{2}\leq Cr^{\frac{2}{m}}$
and (6.37) yields (6.30) for $j=2$. In the same manner, by using Proposition
6.4 with $Z\geq\frac{3}{4}$ and

$\displaystyle|R_{2}[\varphi]-R_{2}[\psi]|\leq|[\varphi-\psi]_{rr}||Z_{\varphi}^{n-1}-1|+|[\psi]_{rr}||Z_{\varphi}^{n-1}-Z_{\psi}^{n-1}|,$
(6.40)

we can show (6.32) for $j=2$.

Next, to show (6.32) for $j=1$, we consider

$g(t)=(1+[\varphi]_{r}+t)^{1-p}-1-(1-p)([\varphi]_{r}+t),$ (6.41)

and apply the Taylor’s theorem so that we can obtain $C(p)$ satisfying

$|g(t)-g(0)-(1-p)((1+[\varphi]_{r})^{-p}-1)t|\leq C|t|^{2}$ (6.42)

for $|t|\leq\frac{1}{2}$. Hence, putting $t=[\psi-\varphi]_{r}$, we get (6.32)
for $j=1$. Note that the proof for (6.32) with $j=3$ is identical. To prove
the last estimate (6.31), we consider

$g(t)=(\hat{Y}_{\varphi,\psi}+t)^{1-p}-(1+[\varphi]_{r}+t)^{1-p},$ (6.43)

where

$\hat{Y}_{\varphi,\psi}:=1+\tfrac{1}{m+1}r+[\varphi]_{r}+r[\psi]_{s}.$ (6.44)

Then, the Taylor remainder theorem says that there is $C(p)$ such that

$|g(t)-g(0)-(1-p)[\hat{Y}_{\varphi,\psi}^{-p}-(1+[\varphi]_{r})^{-p}]\,t\;|\leq
C|t|^{2}$ (6.45)

holds for $|t|\leq\frac{1}{2}$. Observing
$\hat{Y}_{\varphi,\psi},Z_{\varphi}\geq\frac{3}{4}$, we apply Proposition 6.4
to get

$|\hat{Y}_{\varphi,\psi}^{-p}-(1+[\varphi]_{r})^{-p}|\leq Cr.$ (6.46)

Therefore, putting $t=[\psi-\varphi]_{r}$ yields

$|Q[\psi]-g(0)|\leq
C(r+\|\varphi-\psi\|_{C^{1}_{w}(I)})\|\varphi-\psi\|_{C^{1}_{w}(I)}.$ (6.47)

On the other hand, using Proposition 6.4 and
$\hat{Y}_{\varphi,\psi},Y_{\varphi}\geq\frac{3}{4}$, we can obtain

$|g(0)-Q[\varphi]|=|\hat{Y}_{\varphi,\psi}^{1-p}-Y_{\varphi}^{1-p}|\leq
Cr|[\psi-\varphi]_{s}|\leq Cr\|\psi-\varphi\|_{C^{1}_{w}(I)}.$ (6.48)

This completes the proof. ∎

######  Lemma 6.6.

There is some constant $C_{3}$ depending only on $n,p$ with $m>0$ such that if
$|g(r)|\leq r^{\delta}$ then

$\hat{w}(r):=\int_{0}^{r}s^{\frac{1}{m}-1}\int_{0}^{s}m^{\frac{1}{m}-1}g(t)\,\mathrm{d}t\,\mathrm{d}s$
(6.49)

satisfies $\|\hat{w}\|_{C^{2}_{w}((0,r])}\leq C_{3}r^{\delta}$ for $r>0$.

###### Proof.

Since

$m^{1-\frac{1}{m}}\hat{w}_{rr}=r^{\frac{1}{m}-1}g(r)+(\tfrac{1}{m}-1)r^{\frac{1}{m}-2}\int_{0}^{r}g(t)\,\mathrm{d}t,$
(6.50)

we have

$m^{1-\frac{1}{m}}|\hat{w}_{rr}|\leq
r^{\frac{1}{m}-1}|g(r)|+|\tfrac{1}{m}-1|r^{\frac{1}{m}-2}\int_{0}^{r}|g(t)|\,\mathrm{d}t\leq
Cr^{\frac{1}{m}-1+\delta}.$ (6.51)

Similarly, we can compute $|\hat{w}|$ and $|\hat{w}_{r}|$ to obtain the
desired result. ∎

###### Proof of Theorem 6.2.

We recall the constants $C_{1},C_{2},C_{3},\delta$ in the above lemmas and we
choose $r_{0}\in(0,1]$ satisfying

$\displaystyle
8C_{1}C_{3}r_{0}^{\delta}+8C_{2}C_{3}(1+4C_{1}C_{3})r_{0}^{\delta}\leq 1.$
(6.52)

Then, combining with Lemma 6.3 and Lemma 6.6 yields

$2\|w_{1}\|_{C^{2}_{w}(I)}\leq 2C_{1}C_{3}r^{\delta}\leq\tfrac{1}{4}$ (6.53)

for every $r\leq r_{0}$, where $I=(0,r]$.

Then, we will inductively show that for each $i\in\mathbb{N}$ we have

$\|w_{i}-w_{i-1}\|_{C^{2}_{w}(I)}\leq 2^{-(i-1)}\|w_{1}\|_{C^{2}_{w}(I)},$
(6.54)

and

$\displaystyle\|w_{i}\|_{C^{2}_{w}(I)}\leq(2-2^{-(i-1)})\|w_{1}\|_{C^{2}_{w}(I)}\leq\tfrac{1}{4}.$
(6.55)

First of all, the base case $i=1$ is obvious by $w_{0}=0$ and (6.53).

Next, we assume that the above inequalities hold for $i\leq l$. Then,
remembering (6.55), we can apply Lemma 6.5 to obtain

$|E[w_{l}]-E[w_{l-1}]|\leq
4C_{2}(r^{\delta}+4\|w_{1}\|_{C^{2}_{w}(I)})\|w_{l}-w_{l-1}\|_{C^{2}_{w}(I)}.$
(6.56)

Therefore, (6.53) implies

$|E[w_{l}]-E[w_{l-1}]|\leq
4C_{2}(1+4C_{1}C_{3})\,r^{\delta}\,\|w_{l}-w_{l-1}\|_{C^{2}_{w}(I)}.$ (6.57)

Hence, by Lemma 6.6 and (6.52), we have

$\|w_{l+1}-w_{l}\|_{C^{2}_{w}(I)}\leq\tfrac{1}{2}\|w_{l}-w_{l-1}\|_{C^{2}_{w}(I)}.$
(6.58)

Thus, combining (6.54) with $i=l$ shows (6.54) for $i=l+1$, and therefore we
have (6.55) for $i=l+1$.

To conclude, we add (6.54) for all $i\geq 1$ to get pointwise limits

$\displaystyle\bar{w}^{[0]}(r):=\lim_{i\to+\infty}w_{i}(r),$
$\displaystyle\bar{w}^{[1]}(r):=\lim_{i\to+\infty}\tfrac{d}{dr}w_{i}(r),$
$\displaystyle\bar{w}^{[2]}(r):=\lim_{i\to+\infty}\tfrac{d^{2}}{dr^{2}}w_{i}(r).$
(6.59)

for $r\in(0,r_{0}]$. Moreover, for each $k=1,2$

$\bar{w}^{[k-1]}(r_{1})-\bar{w}^{[k-1]}(r_{2})=\int^{r_{1}}_{r_{2}}\bar{w}^{[k]}(r)dr$
(6.60)

holds whenever $0<r_{1}<r_{2}\leq r_{0}$. Hence, $\bar{w}^{[0]}$ and
$\bar{w}^{[1]}$ are continuous, and thus the fundamental theorem of calculus
(FTC) says $\frac{d}{dr}\bar{w}^{[0]}=\bar{w}^{[1]}$. In addition, the
equation (6.3) says

$h_{r}^{m-1}Z_{\bar{w}^{[0]}}^{n-1}(h_{rr}+\bar{w}^{[2]})=(1+r)^{n-1}X_{\bar{w}^{[0]}}^{\frac{m+2}{2}}Y_{\bar{w}^{[0]}}^{1-p}$
(6.61)

where $X,Y,Z$ are given in (6.33). Hence, $\bar{w}^{[2]}$ is also continuous,
and therefore we have $\frac{d}{dr}\bar{w}^{[1]}=\bar{w}^{[2]}$ by the FTC
again. Thus, $\bar{w}^{[0]}\in C^{2}$ is a solution to (6.17) satisfying

$\|\bar{w}^{[0]}\|_{C^{2}_{w}((0,r])}\leq 2\|w_{1}\|_{C^{2}_{w}((0,r])}\leq
2C_{1}C_{3}r^{\delta}=:C_{0}r^{\delta}$ (6.62)

for every $r\in(0,r_{0}]$ as desired. Note that we may choose smaller $r_{0}$
to have

$C_{0}r_{0}^{\delta}\leq\tfrac{1}{10}\min\\{m,m^{-1}\\}.$ (6.63)

This completes the proof. ∎

###  6.2. Example revisited

In this section, we prove Theorem 1.2 (ii) by revisiting the example provided
by Chou–Wang [33] and Bianchi–Böröczky–Colesanti [6]. While the example
constructed in the previous section has a flat side, the example below does
not.

###### Proof of Theorem 1.2.

For each $p\in(-n+1,1)\cup(1,n+1)$, Chou and Wang [33, Section 6] proved that
there exists a generalized solution of (1.2) that is not smooth even for a
smooth positive function $f$. Let us recall this example and prove that the
associated hypersurface $\Sigma$ is not of $C^{1,1}$ when $p\in(1,n+1)$. Let
$\tilde{u}$ be the restriction of $u$ on the tangent hyperplane of the
$n$-sphere at the south pole. If $\tilde{u}$ is given by
$\tilde{u}(y)=|y|^{2\alpha}$ with $\alpha=n/(n-p+1)$, then it satisfies

$\det D^{2}\tilde{u}=(2\alpha)^{n}(2\alpha-1)\tilde{u}^{p-1}\quad\text{in
}\mathbb{R}^{n}$ (6.64)

in the generalized sense. Thus, $u$ is a generalized solution of (1.2) near
the south pole, with a smooth positive function $f$. Note that $f$ is given by

$f(z)=(2\alpha)^{n}(2\alpha-1)\left(\frac{1}{|z_{n+1}|}\right)^{n+2p}$ (6.65)

near the south pole.

We define

$\bar{v}(r)=cr^{\frac{2\alpha}{2\alpha-1}}=cr^{\frac{2n}{n+p-1}},\quad
c=\frac{2\alpha-1}{(2\alpha)^{2\alpha/(2\alpha-1)}},$ (6.66)

and $v(x)=\bar{v}(|x|)$, $x\in\mathbb{R}^{n}$. Then, $\Sigma$ is represented
by $\\{(x,v(x))\\}$ near the south pole. Indeed, we have for $y=Dv(x)$

$\tilde{u}(y)=|y|^{2\alpha}=(2\alpha)^{-\frac{2\alpha}{2\alpha-1}}|x|^{\frac{2\alpha}{2\alpha-1}}=|x|\bar{v}(|x|)-\bar{v}(|x|)=\langle(x,v(x)),(y,-1)\rangle.$
(6.67)

We observe that $v$ is of $C^{1,\frac{n-p+1}{n+p-1}}$, not $C^{1,1}$, provided
$p\in(1,n+1)$. ∎

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