# Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant

Jurgen Julio-Batalla  Universidad Industrial de Santander, Carrera 27 calle 9,
680002, Bucaramanga, Santander, Colombia<EMAIL_ADDRESS>
###### Abstract.

We consider on a closed Riemannian spin manifold $(M^{n},g,\sigma)$ the
spinorial Yamabe type equation
$D_{g}\varphi=\lambda|\varphi|^{\frac{2}{n-1}}\varphi$, where $\varphi$ is a
spinor field and $\lambda$ is a positive constant. For a normalized solution
$\varphi$ of this equation we find a positive lower bound for $\lambda^{2}$.
As an application we obtain an explicit lower bound of the Bär-Hijazi-Lott
invariant for some spin manifolds with positive scalar curvature.

The author was supported by project 3756 of Vicerrectoría de Investigación y
Extensión of Universidad Industrial de Santander

##  1\. Introduction

Let $(M^{n},g,\sigma)$ be a closed Riemannian spin manifold with a spin
structure $\sigma:P_{spin}(M)\rightarrow P_{SO}(M)$. We consider the
representation $\rho:spin(n)\rightarrow Aut(S_{n})$ of the spin group
compatible with Clifford multiplication and the complex vector bundle
$SM=P_{spin}(M)\times_{\rho}S_{n}$ associated to the principal bundle
$\sigma:P_{spin}(M)\rightarrow P_{SO}(M)$. This vector bundle is known as the
spinor bundle and it is endowed with a natural spinorial Levi-Civita
connection $\nabla$, a pointwise Hermitian scalar product
$\langle\cdot,\cdot\rangle$ and an elliptic differential operator $D_{g}$
(called Dirac operator).

In this paper, we are interested in the spinorial Yamabe-type equation

(1.1) $D_{g}\varphi=\lambda|\varphi|^{\frac{2}{n-1}}\varphi,$

where $\varphi\in\Gamma(SM)$, $\lambda\in\mathbb{R}$ and the scalar curvature
of the metric $g$, $sc_{g}$, is positive.

When $\lambda=1$ this equation is referred as the spinorial Yamabe equation
([2];[3];[4]). Solutions of this equation are related to some interesting
geometric problems. For instance when $n=2$ solutions of the spinorial Yamabe
equation produce conformally immersed constant mean curvature surfaces in
$\mathbb{R}^{3}$ ([7, 2]), and when $n\geq 2$ it is related to conformally
immersed constant mean curvature hypersurfaces in $\mathbb{R}^{n+1}$. Also
there is an interesting relation with a conformal spinorial invariant. Indeed,
in his Habilitation [2] B. Ammann introduced the Bär-Hijazi-Lott invariant
$\lambda_{min}^{+}(M^{n},[g],\sigma)$ of the spin manifold $(M,g,\sigma)$
defined by

$\lambda_{min}^{+}(M^{n},[g],\sigma):=\inf\limits_{g^{\prime}\in[g]}\lambda_{1}(D_{g^{\prime}})vol(M,g^{\prime})^{1/n},$

where $\lambda_{1}(D_{g^{\prime}})$ is the first positive eigenvalue of Dirac
operator $D_{g^{\prime}}$ with respect to the conformal metric $g^{\prime}$
and $vol(M,g^{\prime})$ is the volume of $M$ with respect to the metric
$g^{\prime}$.

J. Lott proved in [11] that this invariant is positive when the Dirac operator
$D_{g^{\prime}}$ (for some conformal metric $g^{\prime}$) is invertible. Later
B. Ammann in ([1]) extended the result to any closed spin Riemannian manifold.

In order to obtain generalized metrics which realize the Bär-Hijazi-Lott
invariant, B. Ammann proved ([2]) that this problem is equivalent to the
existence of a spinor field $\varphi$ minimizing the Rayleigh type quotient

(1.2)
$J(\psi):=\dfrac{\left(\int_{M}|D_{g}\psi|^{\frac{2n}{n+1}}dv_{g}\right)^{(n+1)/n}}{|\int_{M}\langle
D_{g}\psi,\psi\rangle dv_{g}|}$

on the space of smooth spinor fields. Here $dv_{g}$ is the volume element of
$(M^{n},g)$.

Moreover, the infimum of $J$ is $\lambda_{min}^{+}(M^{n},[g],\sigma)$ and the
corresponding Euler-Lagrange equation is

$D_{g}\varphi=|\varphi|^{\frac{2}{n-1}}\varphi\quad\text{with}\quad\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_{g}=\lambda_{min}^{+}(M^{n},[g],\sigma)^{n}$

or equivalently

(1.3)
$D_{g}\varphi=\lambda_{min}^{+}(M^{n},[g],\sigma)|\varphi|^{\frac{2}{n-1}}\varphi\hskip
28.45274pt\text{with}\hskip
28.45274pt\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_{g}=1.$

From this discussion, let us point out here that solutions of equation (1.1)
normalized by
$\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_{g}=1$ can be related to the Bär-Hijazi-
Lott invariant. With this in mind, we state the main result of this note as
follows

######  Theorem 1.1.

Let $(M^{n},g,\sigma)$ be a closed Riemannian spin manifold of positive scalar
curvature. Assume $n\geq 3$ and there is an open bounded domain $\Omega\subset
M$ such that $(\Omega,g)$ is conformally flat. If $\varphi$ is a solution of
the equation (1.1) with $\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_{g}=1$. Then

$\lambda^{2}\geq\dfrac{n^{2}}{4}\dfrac{(n-2)}{(n-1)}\omega_{n}^{2/n},$

where $\omega_{n}$ is the volume of the round sphere $(\mathbb{S}^{n},g_{0})$.

The general existence of solutions for equation (1.1) is a very difficult
task. For instance, a standard variational approach can not produce nontrivial
solutions because $2n/(n+1)$ is the critical exponent in the Sobolev
inequality of the spaces involved in the definition of the functional (1.2).

However there exists a criterion (similar to the Yamabe problem) to provide
solutions to this problem. In fact if the Bär-Hijazi-Lott invariant satisfies

(1.4) $\lambda_{min}^{+}(M^{n},[g],\sigma)<\frac{n}{2}\omega_{n}^{1/n}.$

Then there is a nontrivial spinor solution of (1.3) and therefore there is a
generalized metric that attain the invariant
$\lambda_{min}^{+}(M^{n},[g],\sigma)$ (see [2] for more details).

It is well-known that
$\lambda_{min}^{+}(M^{n},[g],\sigma)\leq\frac{n}{2}\omega_{n}^{1/n}$ for all
closed spin manifolds $(M^{n},g,\sigma)$ (see [4]). Furthermore the equality
is attained by the round sphere of dimension $n$ i.e.,
$\lambda_{min}^{+}(\mathbb{S}^{n},[g_{0}],\sigma_{0})=(n/2)\omega_{n}^{1/n}$,
where $\sigma_{0}$ is the unique spin structure on $\mathbb{S}^{n}$.
Nevertheless it is an open problem to determine when a closed spin manifold
non conformally equivalent to the round sphere satisfies or not the strict
inequality (1.4).

There are very few cases where (1.4) it is known. For example, in the locally
conformally flat setting B. Ammann, E. Humbert and B. Morel give a sufficient
condition to get (1.4). Specifically, if $(M^{n},g,\sigma)$ is locally
conformally flat with invertible Dirac operator $D_{g}$ and the mass-
endomorphism possesses a positive eigenvalue at a point, then the inequality
(1.4) holds (see [3] for more details about this result). In a recent paper
[9] T. Isobe and T. Xu constructed a family of metrics $g(\epsilon)$ on sphere
$\mathbb{S}^{n}$ such that

$\lambda_{min}^{+}(\mathbb{S}^{n},[g(\epsilon)],\sigma_{0})<\frac{n}{2}\omega_{n}^{1/n}$

for $n\geq 4$ and $\epsilon>0$ small enough. The metrics $g(\epsilon)$
restricted to some open domain $\Omega\subset\mathbb{S}^{n}$ are equal to the
round metric $g_{0}$ but are not locally conformally flat. For more about the
problem of strict inequality (1.4) see for instance [13].

The manifolds described above not only admit a nontrivial solution of (1.3),
also contain an open domain $\Omega$ for which $(\Omega,g)$ is conformally
flat. Then the Theorem (1.1) implies that

######  Corollary 1.2.

Let $(M^{n},g,\sigma)$ be a closed spin manifold of dimension $n\geq 3$ with
positive scalar curvature. Assume that

  * •

$(M^{n},g)$ is locally conformally flat with a positive eigenvalue for the
mass endomorphism at some point or,

  * •

$(M^{n},g,\sigma)=(\mathbb{S}^{n},g(\epsilon),\sigma_{0})$ for small
$\epsilon>0$ and $g(\epsilon)$ as above.

Then the Bär-Hijazi-Lott invariant satisfies that

$\lambda_{min}^{+}(M^{n},[g],\sigma)^{2}\in\left(\dfrac{(n-2)}{8(n-1)}n^{2}\omega_{n}^{2/n},\frac{1}{4}n^{2}\omega_{n}^{2/n}\right).$

There is another invariant for which our estimation applies. Let $C(M)$ be the
set of conformal classes of metrics on closed spin manifold $M^{n}$. The
$\tau-$invariant of $M^{n}$ with spin structure $\sigma$ is defined by

$\tau(M^{n},\sigma):=\sup\limits_{[g]\in
C(M)}\lambda^{+}_{min}(M^{n},[g],\sigma).$

Clearly the $\tau-$invariant for manifolds in Corollary 1.2 satisfies

$\left(\dfrac{\tau(M^{n},\sigma)}{n\omega_{n}^{1/n}}\right)^{2}\in\left(\dfrac{(n-2)}{8(n-1)},\frac{1}{4}\right].$

In the literature there are other type of lower bounds for the Bär-Hijazi-Lott
invariant. In dimension $n\geq 3$, O. Hijazi in [8] proved that

$\lambda_{min}^{+}(M^{n},[g],\sigma)^{2}\geq\frac{n}{4(n-1)}Y(M^{n},[g]),$

where $Y(M^{n},[g])$ is the Yamabe constant of the conformal class $[g]$ on
$M$.

This inequality exhibits a strong connection with the Yamabe problem. For
instance the Hijazi’s inequality provides a spinorial proof of the Yamabe
problem when the inequality (1.4) holds.

Therefore, estimations of the invariant $\lambda_{min}^{+}(M^{n},[g],\sigma)$
can help to estimate the Yamabe constant from a spinorial point of view.

The plan of this note is the following. In section 2 we will recall some
background material on spin manifolds and the equation (1.1). In section 3 we
shall show all ingredients needed later, and in section 4 we deal with the
proof of the main theorem.

##  2\. Preliminaries

In this section we recall some basic facts about spin manifolds and Dirac
operator. Also some results for solutions to Dirac-type equations are
recalled. For more details see ([2],[6]).

On a closed oriented Riemannian manifold $(M^{n},g)$ we can define a
$SO(n)$-principal bundle $P_{SO}M$ over $M$ of oriented $g-$orthonormal bases
at $x\in M$. Let $f_{\alpha,\beta}$ be the transition functions. For $n\geq 3$
there exists the universal covering $\sigma:spin(n)\rightarrow SO(n)$ where
$spin(n)$ is the group generated by even unit-length vector of
$\mathbb{R}^{n}$ in the real Clifford algebra $Cl_{n}$ (the associative
$\mathbb{R}-$algebra generated by relation $VW+WV=-2(V,W)$ for Euclidean
metric $(,)$). The manifold $M$ is called spin if there is a
$spin(n)-$principal bundle $P_{spin}M$ over $M$ such that it is a double
covering of $P_{SO}M$ whose restriction to each fiber is
$\sigma:spin(n)\rightarrow SO(n)$. This double covering from $P_{spin}M$ to
$P_{SO}M$, $\sigma$, it is known as spin structure.

There are four special structures associated to a spin manifold
$(M^{n},g,\sigma)$:

  1. (1)

A complex vector bundle $SM:=P_{spin}(M)\times_{\rho}S_{n}$ where
$\rho:spin(n)\rightarrow Aut(S_{n})$ is the restriction to $spin(n)$ of an
irreducible representation $\rho:\mathbb{C}l_{n}\rightarrow End(S_{n})$ of the
complex Clifford algebra $\mathbb{C}l_{n}\simeq
Cl_{n}\otimes_{\mathbb{R}}\mathbb{C}$, $S_{n}\simeq\mathbb{C}^{N}$ and
$N=2^{[n/2]}$.

  2. (2)

The Clifford multiplication $m$ on $SM$ defined by

$\displaystyle m:TM\times SM$ $\displaystyle\rightarrow SM$ $\displaystyle
X\otimes\varphi$ $\displaystyle\mapsto X\cdot_{g}\varphi:=\rho(X)\varphi.$

  3. (3)

A Hermitian product $\langle\cdot,\cdot\rangle$ on sections of $SM$.

  4. (4)

A Levi-Civita connection $\nabla$ on $SM$.

All these structures are compatible in the following sense:

$\displaystyle\langle X\cdot\varphi,\psi\rangle$
$\displaystyle=-\langle\varphi,X\cdot\psi\rangle,$ $\displaystyle
X(\langle\varphi,\psi\rangle)$
$\displaystyle=\langle\nabla_{X}\varphi,\psi\rangle+\langle\varphi,\nabla_{X}\psi\rangle,$
$\displaystyle\nabla_{X}(Y\cdot\varphi)$
$\displaystyle=\nabla_{X}Y\varphi+Y\cdot\nabla_{X}\varphi,$

for all $X,Y\in\Gamma(TM)$ and $\varphi,\psi\in\Gamma(SM)$.

On the other hand, on $\Gamma(SM)$ we have a connection Laplacian
$\nabla^{*}\nabla$ where
$\nabla^{*}:\Gamma(Hom(TM,SM))\rightarrow\Gamma(SM)$ is the adjoint of the
Levi-Civita connection $\nabla:\Gamma(SM)\rightarrow\Gamma(Hom(TM,SM))$
defined implicitly by

$\int_{M}\langle\nabla^{*}F,\varphi\rangle dv_{g}=\int_{M}\langle
F,\nabla\varphi\rangle dv_{g}.$

This Laplacian satisfies a very useful Bochner-type formula. For spinor field
$\varphi$ compactly supported it is known that

(2.1)
$\Delta\left(\frac{|\varphi|^{2}}{2}\right)=\langle\nabla\varphi,\nabla\varphi\rangle-\langle\nabla^{*}\nabla\varphi,\varphi\rangle,$

where $\Delta$ is the non-positive Laplacian-Beltrami on $(M^{n},g)$.

Now identifying $\Gamma(Hom(TM,SM))$ as $\Gamma(TM\otimes SM)$ we can define
the Dirac operator $D_{g}$ as the composition of $\nabla$ with the Clifford
multiplication $m$ i.e. $D_{g}:=m\circ\nabla$. For a local orthonormal frame
$\\{E_{i}\\}$ we have

$D_{g}\varphi=\sum\limits_{i=1}^{n}E_{i}\cdot_{g}\nabla_{E_{i}}\varphi.$

For our purpose we just recall the following properties of Dirac operator

  1. (1)

(Green’s formula) For an open domain with smooth boundary $\Omega\subset M$,

$\int_{\Omega}\langle D_{g}\varphi,\psi\rangle
dv_{g}=\int_{\Omega}\langle\varphi,D_{g}\psi\rangle
dv_{g}+\int_{\partial\Omega}\langle\varphi,Z\cdot\psi\rangle dv_{g}^{n-1},$

where $Z$ is a unit normal vector field on $\partial\Omega$ and $dv_{g}^{n-1}$
is the volume element of $\partial\Omega$.

  2. (2)

(Schrödinger-Lichnerowicz formula)

$D_{g}^{2}=\nabla^{*}\nabla+\frac{1}{4}sc_{g}Id_{\Gamma(SM)}.$

  3. (3)

(Penrose operator) The composition of the Levi-Civita connection $\nabla$ with
the orthogonal projection on the kernel of the Clifford multiplication $m$
defines an operator $P$ known as Penrose operator. This operator satisfies
that

(2.2) $|\nabla\varphi|^{2}=\frac{1}{n}|D_{g}\varphi|^{2}+|P\varphi|^{2}.$

We finish this section with a couple of facts about the solutions of equation
1.1. These can be consult in Ammann’s Habilitation [2].

######  Remark 2.1.

  * •

Due to the possible existence of nodal sets for spinor fields $\varphi$
solutions of 1.1 by regularity theory we have that $\varphi\in
C^{1,\alpha}(SM)\cap C^{\infty}(S(M-\\{\varphi^{-1}(0)\\})).$

  * •

The equation 1.1 satisfies a weak unique continuation property. This means
that the nodal set of a $C^{1}$ solution does not contain any nonempty open
set [2, Corollary 4.5.2].

##  3\. Estimation of Parameters

In this section we will prove some key results.

We consider the Dirac-type equation

(3.1) $D_{g}\varphi=\lambda|\varphi|^{q}\varphi$

where $\lambda,q\in\mathbb{R}_{+}$.

######  Lemma 3.1.

Let $\varphi$ be a solution of (3.1) without zeros in an open domain
$\Omega\subset M$. Assume $\varphi\in C^{\infty}(S\Omega)$. For any $u\in
C_{0}^{2}(\Omega)$ it satisfies the following integral identity

(3.2) $0=\int_{\Omega}|\varphi|^{2}\left\\{\frac{-\Delta
u}{2}+\frac{u}{4}sc_{g}+\left(\frac{u}{n}-u\right)\lambda^{2}|\varphi|^{2q}\right\\}dv_{g}+\int_{\Omega}u|P\varphi|^{2}dv_{g}.$

###### Proof.

We assume a nontrivial function $u\in C^{2}_{0}(\Omega)$. Applying the
Bochner-type formula (2.1) to $\varphi$, multiplying by $u$ and integrating we
get

$0=-\int_{\Omega}\frac{u}{2}\Delta(|\varphi|^{2})dv_{g}+\int_{\Omega}u|\nabla\varphi|^{2}dv_{g}-\int_{\Omega}u\langle\nabla^{*}\nabla\varphi,\varphi\rangle
dv_{g}.$

By the Schrödinger-Lichnerowicz formula it follows that

(3.3)
$0=-\int_{\Omega}\frac{u}{2}\Delta(|\varphi|^{2})dv_{g}+\int_{\Omega}u|\nabla\varphi|^{2}dv_{g}-\int_{\Omega}u\langle
D_{g}^{2}\varphi,\varphi\rangle
dv_{g}+\frac{1}{4}\int_{\Omega}u\;sc_{g}|\varphi|^{2}dv_{g}.$

Using Green’s formula for Dirac operator we have,

$\displaystyle\int_{\Omega}u\langle D_{g}^{2}\varphi,\varphi\rangle dv_{g}$
$\displaystyle=\int_{\Omega}\langle D_{g}\varphi,D_{g}(u\varphi)\rangle
dv_{g}+\int_{\partial\Omega}u\langle D_{g}\varphi,N\cdot\varphi\rangle
dv^{n-1}_{g}$ $\displaystyle=\int_{\Omega}\left(\langle
D_{g}\varphi,uD_{g}\varphi\rangle+\langle D_{g}\varphi,\nabla
u\cdot\varphi\rangle\right)dv_{g}+\int_{\partial\Omega}u\langle
D_{g}\varphi,N\cdot\varphi\rangle dv^{n-1}_{g}$
$\displaystyle=\int_{\Omega}u\lambda^{2}|\varphi|^{2q}|\varphi|^{2}dv_{g}+\int_{\Omega}\lambda|\varphi|^{q}\langle\varphi,\nabla
u\cdot\varphi\rangle dv_{g}+\int_{\partial\Omega}u\langle
D_{g}\varphi,N\cdot\varphi\rangle dv^{n-1}_{g}.$

In last equation we used that $\varphi$ is a solution of equation (3.1).

On the other hand,

$\displaystyle\int_{\Omega}-\frac{u}{2}\Delta(|\varphi|^{2})dv_{g}=\int_{\Omega}-\frac{\Delta
u}{2}|\varphi|^{2}dv_{g}+\int_{\partial\Omega}\left(\frac{|\varphi|^{2}}{2}\langle\nabla
u,N\rangle-\frac{u}{2}\langle N,\nabla|\varphi|^{2}\rangle\right)dv_{g}.$

Since that $supp\;(u)\subset\Omega$, $u=0$ and $\nabla u=0$ on
$\partial\Omega$. Therefore all previous boundary terms vanish i.e.

(3.4) $\displaystyle\int_{\Omega}u\langle D_{g}^{2}\varphi,\varphi\rangle
dv_{g}$
$\displaystyle=\int_{\Omega}u\lambda^{2}|\varphi|^{2q}|\varphi|^{2}dv_{g}+\int_{\Omega}\lambda|\varphi|^{q}\langle\varphi,\nabla
u\cdot\varphi\rangle dv_{g}$ (3.5)
$\displaystyle\int_{\Omega}-\frac{u}{2}\Delta(|\varphi|^{2})dv_{g}$
$\displaystyle=\int_{\Omega}-\frac{\Delta u}{2}|\varphi|^{2}dv_{g}$

Using the spliting (2.2) and combining the relations (3.4);(3.5) into the
equation (3.3) we obtain,

$\displaystyle 0$
$\displaystyle=-\int_{\Omega}\frac{u}{2}\Delta(|\varphi|^{2})dv_{g}+\int_{\Omega}u|\nabla\varphi|^{2}dv_{g}-\int_{\Omega}u\langle
D_{g}^{2}\varphi,\varphi\rangle
dv_{g}+\frac{1}{4}\int_{\Omega}u\;sc_{g}|\varphi|^{2}dv_{g}$
$\displaystyle=-\int_{\Omega}\frac{\Delta
u}{2}|\varphi|^{2}dv_{g}+\int_{\Omega}u|P\varphi|^{2}dv_{g}+\int_{\Omega}\frac{u}{n}|D_{g}\varphi|^{2}dv_{g}+\frac{1}{4}\int_{\Omega}u\;sc_{g}|\varphi|^{2}dv_{g}$
$\displaystyle-\int_{\Omega}u\lambda^{2}|\varphi|^{2q}|\varphi|^{2}dv_{g}-\int_{\Omega}\lambda|\varphi|^{q}\langle\varphi,\nabla
u\cdot\varphi\rangle dv_{g}$
$\displaystyle=\int_{\Omega}|\varphi|^{2}\left\\{\frac{-\Delta
u}{2}+\frac{u}{4}sc_{g}+\left(\frac{u}{n}-u\right)\lambda^{2}|\varphi|^{2q}\right\\}dv_{g}$
$\displaystyle+\int_{\Omega}u|P\varphi|^{2}dv_{g}-\int_{\Omega}\lambda|\varphi|^{q}\langle\varphi,\nabla
u\cdot\varphi\rangle dv_{g}.$

Now note that $\langle\varphi,\nabla u\cdot\varphi\rangle$ is a pure
imaginary:

$\langle\varphi,\nabla u\cdot\varphi\rangle=-\langle\nabla
u\cdot\varphi,\varphi\rangle=-\overline{\langle\varphi,\nabla
u\cdot\varphi\rangle}.$

Hence the identity (3.2) follows.

∎

In the next, we will find conditions for which the right-hand side in (3.2) is
non-negative. For that purpose, we define $L(u):=-2\Delta u+sc_{g}u$ and
$f(x):=4\left(1-\frac{1}{n}\right)|\varphi|^{2q}(x)$, where $\varphi$ is (as
in the previous Lemma 3.1) a solution of (3.1) such that $\varphi\in
C^{\infty}(S\Omega)$ and $\Omega\cap\varphi^{-1}(0)=\emptyset$.

We consider a smooth open bounded domain $\Omega^{\prime}\subset\Omega\subset
M$ and the weighted eigenvalue problem

(3.6) $Lu=\lambda^{2}f(x)u,\quad x\in\Omega^{\prime},$

with Dirichlet boundary condition $u=0$ on $\partial\Omega^{\prime}$.

######  Lemma 3.2.

There exists a $\lambda_{1}(f)>0$ such that

  1. (1)

For $\lambda^{2}=\lambda_{1}(f)$ there is a positive solution of problem
(3.6);

  2. (2)

for $\lambda^{2}<\lambda_{1}(f)$ there is a positive function $u_{\lambda}$ in
$\Omega^{\prime}$ such that $L(u_{\lambda})-\lambda^{2}f(x)u_{\lambda}\in
C^{\infty}_{+}(\Omega^{\prime})$ and $u_{\lambda}=0$ on
$\partial\Omega^{\prime}.$

###### Proof.

We use the standard Fredholm theory.

Since $sc_{g}>0$ the operator $L$ is uniformly elliptic and self-adjoint in
$H^{1}_{0}(\Omega)$ (the closure of the space
$C^{\infty}_{0}(\Omega^{\prime})$ in $L^{2}_{1}(\Omega^{\prime})$). Let
$K:L^{2}(\Omega^{\prime})\rightarrow H^{1}_{0}(\Omega^{\prime})$ the Green
operator associated with $L$, i.e. $K(h)=u,$ where $u$ is the unique weak
solution of

$\displaystyle L(u)$ $\displaystyle=h\quad\text{in}\;\Omega^{\prime},$
$\displaystyle u$ $\displaystyle=0\quad\text{on}\;\partial\Omega^{\prime}.$

We note that, by Sobolev embedding and $L^{p}$ theory, this operator is
compact as a map from $H^{1}_{0}(\Omega^{\prime})$ into itself.

For all $u\in H^{1}_{0}(\Omega^{\prime})$ we have that $fu\in
H^{1}_{0}(\Omega^{\prime})$ ($f\in C^{\infty}(\Omega^{\prime})$). Therefore,
we can define the compact linear operator
$T:H^{1}_{0}(\Omega^{\prime})\rightarrow H^{1}_{0}(\Omega^{\prime})$ by
$T(u)=K(fu)$.

The Riesz-Fredholm theory allows us to obtain a sequence
$(\mu_{i}(f,\Omega^{\prime}))$ of non-increasing values such that

  * •

$\mu_{i}\rightarrow 0$ as $i\rightarrow\infty$.

  * •

$Ker(\mu_{i}Id-T)\neq\emptyset\quad\forall i=1,2,...$

  * •

$\mu Id-T$ is bijective for $\mu\neq\mu_{i}$.

Since $f\in C^{\infty}(\Omega^{\prime})$ the eigenfunctions $u_{i}$ associated
to eigenvalues $\lambda_{i}(f,\Omega^{\prime})=1/\mu_{i}(f,\Omega^{\prime})$
are classical solutions of (3.6).

Moreover, the first eigenvalue
$\lambda_{1}(f):=\lambda_{1}(f,\Omega^{\prime})=\frac{1}{\mu_{1}(f,\Omega^{\prime})}$
of the weighted problem (3.6) satisfies that is simple and its eigenspace is
generated by a positive eigenfunction $u_{1}\in H_{0}^{1}(\Omega)$. Therefore
(1) follows.

On the other hand we get (2) applying the weak maximum principle: For a
positive smooth function $h$ on $M$ and $\lambda^{2}<\lambda_{1}(f)$

$\displaystyle L(u)$
$\displaystyle=\lambda^{2}fu+h,\quad\text{in}\quad\Omega^{\prime},$
$\displaystyle u$ $\displaystyle=0\hskip
54.06006pt\text{on}\quad\Omega^{\prime}$

has a positive solution $u_{\lambda}$. ∎

From the identity in Lemma 3.1 and Lemma 3.2 we obtain some restrictions on
solutions of the equation (3.1). Indeed

######  Theorem 3.3.

Let $\varphi$ be a spinor field solution of (3.1). Assume that $\varphi\in
C^{2}(S\Omega)$ for some open domain $\Omega\subset M$ such that
$\Omega\cap\varphi^{-1}(0)=\emptyset$. Let $\Omega^{\prime}\subset\Omega$ be a
smooth open domain. Then

$\lambda^{2}\geq\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{2q},\Omega^{\prime}\right),$

where $\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{2q},\Omega^{\prime}\right)$
is the first positive eigenvalue of problem (3.6) with Dirichlet boundary
condition.

###### Proof.

Suppose
$\lambda^{2}<\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{2q},\Omega^{\prime}\right)$.
By (2) in Lemma 3.2 we have a function $u_{\lambda}\in
C^{2}_{+}(\Omega^{\prime})$ such that
$L(u_{\lambda})-\lambda^{2}fu_{\lambda}(\in C^{\infty}(\Omega^{\prime}))$ is
strictly positive in $\Omega^{\prime}$ and $u_{\lambda}=0$ on
$\partial\Omega^{\prime}$.

We define $u_{\lambda}=0$ in $\Omega-\Omega^{\prime}$.

From identity in Lemma 3.1,

$\displaystyle 0$
$\displaystyle=\int_{\Omega}|\varphi|^{2}\left\\{\frac{-\Delta
u_{\lambda}}{2}+\frac{u_{\lambda}}{4}sc_{g}+\left(\frac{u_{\lambda}}{n}-u_{\lambda}\right)\lambda^{2}|\varphi|^{2q}\right\\}dv_{g}+\int_{\Omega}u_{\lambda}|P\varphi|^{2}dv_{g}$
$\displaystyle 0$
$\displaystyle=\int_{\Omega}|\varphi|^{2}\\{L(u_{\lambda})-\lambda^{2}fu_{\lambda}\\}dv_{g}+\int_{\Omega}4u_{\lambda}|P\varphi|^{2}dv_{g}$
$\displaystyle 0$
$\displaystyle=\int_{\Omega^{\prime}}|\varphi|^{2}\\{L(u_{\lambda})-\lambda^{2}fu_{\lambda}\\}dv_{g}+\int_{\Omega^{\prime}}4u_{\lambda}|P\varphi|^{2}dv_{g}.$

Hence $\varphi=0$ in $\Omega^{\prime}$ and this is a contradiction.

∎

######  Remark 3.4.

Similarly to the proof of the previous Theorem, the case
$\lambda^{2}=\lambda_{1}\left(\frac{4(n-1)|\varphi|^{2q}}{n},\Omega^{\prime}\right)$
implies that $P\varphi=0$ in $\Omega^{\prime}$.

We highlight that the equation $P\varphi=0$ is completely understood when
$\Omega^{\prime}=M$. More precisely

  * •

(Lichnerowicz,[10])($n\geq 2$) If the zero set of $\varphi$ is non-empty, then
$(M^{n},g,\sigma)$ is conformally equivalent to
$(\mathbb{S}^{n},g_{0},\sigma_{0})$.

Moreover, by classification of manifolds with Killing spinors and ([5])

  * •

($n\geq 3$) If the zero set of $\varphi$ is empty, then $(M^{n},g)$ is
conformal to an Einstein manifold.

##  4\. Explicit lower bound

In this section we will prove the Theorem (1.1). To give so we shall see
explicit lower bounds for
$\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega^{\prime}\right)$
in a particular family of manifolds.

###### Proof of Theorem (1.1).

First, we recall a well-known result that will be useful in the proof.

######  Lemma 4.1.

[Proposition 2.2, [12]] Let $(M^{n},g)$ be a Riemannian manifold, $n\geq 3$.
Suppose there is a domain $\Omega\subset M$ such that there exists a conformal
immersion from $(\Omega,g)$ into $(\mathbb{S}^{n},g_{0})$. For all $u\in
C^{\infty}(M)$ with $supp(u)\subset\Omega$,

$\left(\int_{M}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}\leq\dfrac{4}{n(n-2)\omega_{n}^{2/n}}\left(\int_{M}|\nabla
u|^{2}dv_{g}+\dfrac{n-2}{4(n-1)}\int_{M}sc_{g}u^{2}dv_{g}\right),$

where $\omega_{n}$ is the volume of the round sphere $(\mathbb{S}^{n},g_{0})$.

Let $\Omega\subset M$ be an open bounded domain such that $(\Omega,g)$ is
conformally flat. We consider a solution $\varphi$ of the equation 1.1 with
$\int_{M}|\varphi|^{\frac{2n}{n-1}}dv_{g}=1$.

From the variational characterization of eigenvalues

$\dfrac{1}{\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega\right)}=\frac{4(n-1)}{n}\sup\left\\{\int_{\Omega}|\varphi|^{4/(n-1)}u^{2}dv_{g}/\int_{\Omega}uLudv_{g}=1\right\\}.$

Using Hölder inequality we get that,

$\int_{\Omega}|\varphi|^{4/(n-1)}u^{2}dv_{g}\leq\left(\int_{\Omega}|\varphi|^{2n/(n-1)}dv_{g}\right)^{2/n}\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}.$

The normalization of $\varphi$ implies that

$\dfrac{1}{\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega\right)}\leq\frac{4(n-1)}{n}\sup\left\\{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}/\int_{\Omega}uLudv_{g}=1\right\\}.$

Hence

$\frac{n}{4(n-1)}\inf\left\\{\dfrac{1}{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}}\quad/\int_{\Omega}uLudv_{g}=1\right\\}\leq\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega\right).$

On the other hand,

$\displaystyle\inf\left\\{\dfrac{1}{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}}\;/\int_{\Omega}uLudv_{g}=1\right\\}$
$\displaystyle=\inf\limits_{supp(u)\subset\Omega}\dfrac{\int_{\Omega}uLudv_{g}}{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}}$
$\displaystyle=2\inf\limits_{supp(u)\subset\Omega}\dfrac{\int_{\Omega}|\nabla
u|^{2}dv_{g}+\frac{1}{2}\int_{\Omega}sc_{g}u^{2}dv_{g}}{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}}.$

Since that $\frac{n-2}{4(n-1)}<1/2$ we get that

$\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega\right)\geq\frac{2n}{4(n-1)}\inf\limits_{supp(u)\subset\Omega}\dfrac{\int_{\Omega}|\nabla
u|^{2}dv_{g}+\frac{n-2}{4(n-1)}\int_{\Omega}sc_{g}u^{2}dv_{g}}{\left(\int_{\Omega}u^{2n/(n-2)}dv_{g}\right)^{(n-2)/n}}.$

Without loss of generality we can suppose that $\Omega$ is simply connected.
Since $(\Omega,g)$ is conformally flat there exists a conformal immersion from
$(\Omega,g)$ into $(\mathbb{S}^{n},g_{0})$. Then by Lemma 4.1

$\lambda_{1}\left(\frac{4(n-1)}{n}|\varphi|^{4/(n-1)},\Omega\right)\geq\dfrac{n}{4(n-1)}\dfrac{n(n-2)\omega_{n}^{2/n}}{2}=\dfrac{n^{2}}{8}\dfrac{(n-2)}{(n-1)}\omega_{n}^{2/n}.$

Moreover by weak unique continuation property [2, Corollary 4.5.2] we can
assume that the nontrivial solution $\varphi$ does not have zero in $\Omega$.
Therefore Theorem (3.3) implies

$\lambda^{2}\geq\dfrac{n^{2}}{8}\dfrac{(n-2)}{(n-1)}\omega_{n}^{2/n}.$

∎

## References

  * [1] B. Ammann, A spin-conformal lower bound of the first positive Dirac eigenvalue, Differ. Geom. Appl. 18:1 (2003), 21-32.
  * [2] B. Ammann,A variational problem in conformal spin geometry, Habilitationsschrift, Universität Hamburg (2003).
  * [3] B. Ammann, E. Humbert, B. Morel,Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds, Comm. Anal. Geom. 14:1 (2006), 163-182.
  * [4] B. Ammann, J.-F. Grosjean, E. Humbert, B. Morel, A spinorial analogue of Aubin’s inequality, Math. Z. 260 (2008) 127–151.
  * [5] H. Baum, T. Friedrich, R. Grunewald, I. Kath,Twistor and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, Band 124, Teubner-Verlag Stuttgart/Leipzig 1991.
  * [6] J.-P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, S. Moroianu A Spinorial Approach to Riemannian and Conformal Geometry. EMS Monographs in Mathematics (2015).
  * [7] T. Friedrich,On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998) 143–157.
  * [8] O. Hijazi,A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.
  * [9] T. Isobe, T. Xu, Solutions of Spinorial Yamabe-type Problems on $S^{m}$ : Perturbations and Applications, Trans. Amer. Math. Soc. 376 (2023), 6397-6446.
  * [10] A. Lichnerowicz, Killing spinors, twistor-spinors and Hijazi inequality, J. Geom. Phys. 5 (1988), 2–18.
  * [11] J. Lott, Eigenvalue bounds for the Dirac operator, Pac. J. Math. 125:1 (1986), 117-126.
  * [12] R. Schoen, S-T Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47–71, MR 0954421
  * [13] Y. Sire, T. Xu, On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem. ArXiv e-prints(2021).