First Boundary Dirac Eigenvalue and Boundary Capacity Potential

Abstract

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface $\Sigma$ bounding a noncompact domain in a spin asymptotically flat manifold $(M^n,g)$ with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of $\Sigma$ in $(M^n,g)$ yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

Appendices

Appendix A. Mass and Capacity

A smooth, connected, n-dimensional Riemannian manifold $(M^n,g)$ is said to be asymptotically flat (of order p) if there exists a compact subset $K\subset M$ such that $M\setminus K$ is a finite disjoint union of ends $M_k$, each of them being diffeomorphic to ${\mathbb {R}}^n$ minus a closed ball ${\overline{B}}$ by a coordinate chart in which the components of the metric satisfy

$$\begin{aligned} g_{ij} = \delta _{ij} +O_2\big (r^{-p}\big )\quad \text {and}\quad R_g\in L^1(M) \end{aligned}$$

for $i,j=1,\ldots ,n$ and $p>(n-2)/2$. Here, $O_2(r^{-p})$ refers to a real-valued function f such that

$$\begin{aligned} |f(x)|+r|\partial f(x)|+r^2|\partial ^2f(x)|\le Cr^{-p} \end{aligned}$$

as r goes to infinity, for some constant $C>0$ and where $\partial$ is the standard derivative in the Euclidean space. Such a coordinate chart is often referred to as a chart at infinity. In the following, we assume that $k=1$ and the general case can be treated in a similar way.

On an asymptotically flat manifold $(M^n,g)$, the ADM mass is defined by

$$\begin{aligned} m_{ADM}(M,g):=\frac{1}{2(n-1) \omega _{n-1}}\lim _{R\rightarrow +\infty }\sum _{i,j=1}^n\int _{{\mathcal {S}}_R}(g_{ij,i}-g_{ii,j}) \frac{x^j}{r}d{\overline{\mu }}_{{{\mathcal {S}}_R}} \end{aligned}$$

where ${\mathcal {S}}_R$ stands for a coordinate sphere of radius for $R>0$, $d{\overline{\mu }}_{{{\mathcal {S}}_r}}$ its Euclidean Riemannian volume element and $g_{ij,s}$ for the derivative of the metric components in the coordinate chart. Although this definition seems to depend on a particular choice of the coordinates chart, it was proved independently by Bartnik [2] and Chruściel [6] that it is in fact a well-defined geometric invariant.

On asymptotically flat manifolds of order p, the boundary capacity potential $\phi \in C^\infty (M)$ satisfying (1.4) has the following expansion

$$\begin{aligned} \phi (x)=\frac{{\mathcal {C}}_g(\Sigma ,M)}{r^{n-2}}+O_2(r^{-(n-2+p)}) \end{aligned}$$

(A.1)

as $r\rightarrow \infty$ (see [1, Theorem 2.2] for example).

Appendix B. The Riemannian Schwarzschild Manifolds

Here, we recall some standard computations in the Riemannian Schwarzschild manifolds. The Riemannian Schwarzschild manifold of mass $m\in {\mathbb {R}}$ is the Riemannian manifold $({\mathbb {M}}^n_m,g_m)$ where

$$\begin{aligned} {\mathbb {M}}^n_m:= \left\{ \begin{array}{ll} {\mathbb {R}}^n\setminus \{0\} &{} \text { if } m>0\\ {\mathbb {R}}^n &{} \text { if } m=0\\ {\mathbb {R}}^{n}\setminus \big \{r\le \big (|m|/2\big )^{1/(n-2)}\big \} &{} \text { if } m<0 \end{array} \right. \end{aligned}$$

and with metric

$$\begin{aligned} g_m=\left( 1+\frac{m}{2r^{n-2}}\right) ^{\frac{4}{n-2}}\delta \end{aligned}$$

where $r:=|x|$ is the Euclidean radius for $x\in {\mathbb {M}}^n_M$. It is a static manifold in the sense that the Lorentzian manifold

$$\begin{aligned} \left( {\mathfrak {L}}^{n+1}:={\mathbb {R}}\times {\mathbb {M}}^n_m,{\mathfrak {g}}_m:=-N_m^2\,dt^2+g_m\right) \end{aligned}$$

is a spacetime which satisfies the Einstein vacuum equations with zero cosmological constant. Here, $N_m$ denotes the smooth harmonic function given by

$$\begin{aligned} N_m(x)=\left( 1-\frac{m}{2r^{n-2}}\right) \left( 1+\frac{m}{2r^{n-2}}\right) ^{-1} \end{aligned}$$

generally referred to as the lapse function. For $r_0\in (r_*,\infty )$ with $r_*=0$ if $m\ge 0$ and $r_*=(|m|/2)^{1/(n-2)}$, we consider the exterior of the region outside a rotationally symmetric sphere defined by

$$\begin{aligned} {\mathbb {M}}_m^n(r_0):=\Big \{x\in {\mathbb {M}}^n_m\,/\,r\ge r_0\Big \}. \end{aligned}$$

This is an n-dimensional, spin, complete, asymptotically flat manifold with zero scalar curvature and connected inner boundary $\Sigma _{r_0}$ with induced metric $\gamma _{r_0}$ isometric to a round sphere with radius

$$\begin{aligned} r_{g_m,r_0}:=r_0\left( 1+\frac{m}{2r_0^{n-2}}\right) ^{\frac{2}{n-2}} \end{aligned}$$

and constant mean curvature

$$\begin{aligned} H_{g_m,r_0}=\frac{n-1}{r_0}\left( 1-\frac{m}{2r_0^{n-2}}\right) \left( 1+\frac{m}{2r_0^{n-2}}\right) ^{-\frac{n}{n-2}}. \end{aligned}$$

Remark that for $m>0$, the region ${\mathbb {R}}\times {\mathbb {M}}^n_m(r_m)$ with $r_m=(m/2)^{1/(n-2)}$ represents the exterior of a black hole with event horizon at $r=r_m$ in $({\mathfrak {L}}^{n+1},{\mathfrak {g}}_m)$. On the other hand, the boundary capacity potential of $\Sigma _{r_0}$ in $\big ({\mathbb {M}}^n_m,g_m\big )$ can be computed to be

$$\begin{aligned} \phi _{r_0}(x)=\left( 1+\frac{m}{2r_0^{n-2}}\right) \left( 1+\frac{m}{2r^{n-2}}\right) ^{-1}\left( \frac{r_0}{r}\right) ^{n-2}. \end{aligned}$$

Thus, it holds on $\Sigma _{r_0}$ that

$$\begin{aligned} \frac{\partial \phi _{r_0}}{\partial \nu }=-\frac{n-2}{r_0}\left( 1+\frac{m}{2r_0^{n-2}}\right) ^{-\frac{n}{n-2}} \end{aligned}$$

and then

$$\begin{aligned} {\mathcal {C}}_{g_m}\left( \Sigma _{r_0},{\mathbb {M}}^n_m(r_0)\right) =\frac{m}{2}+r_0^{n-2}. \end{aligned}$$

It is also relevant to note that

$$\begin{aligned} -2\frac{n-1}{n-2}\frac{\partial \phi _{r_0}}{\partial \nu }-H_{g_m,r_0}=\frac{n-1}{r_{g_m,r_0}} \end{aligned}$$

so that this leads to the following interpretation

$$\begin{aligned} \lambda _1(D\!\!\!\!/\,_{r_0})=-\frac{n-1}{n-2}\frac{\partial \phi _{r_0}}{\partial \nu }-\frac{H_{g_m,r_0}}{2} \end{aligned}$$

in terms of $\lambda _1(D\!\!\!\!/\,_{r_0})$ the first eigenvalue of the Dirac operator $D\!\!\!\!/\,_{r_0}$ of $(\Sigma _{r_0},\gamma _{r_0})$. This implies that the manifold ${\mathbb {M}}_m^n(r_0)$ satisfies the equality case in the estimate of Theorem 1.3.