平行旋量场存在的全息原理和Shi-Tam型不等式
A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type
摘要：
设$Σ=\partial M$是具有非负数量曲率的连通紧致黎曼自旋流形$( M,\langle\;,\;\rangle)$的$n$维边界，且$Σ$的（向内）平均曲率$H$为正.证明了共形度规$\langle\;,\;\rangle_H=H^2\langle\;,\;\rangle$对应的边界上Dirac算子的第一本征值至少为$n/2$，且等式成立当且仅当$ M$上存在平行旋量场.因此，如果$Σ$允许一个具有平均曲率$H_0$的等距和同位旋浸入$ϕ$作为超曲面进入另一个允许平行旋量场的自旋黎曼流形$ M_0$，则\begin{equation} \label{HoloIneq} \int_ΣH\,dΣ\le \int_Σ\frac{H^2_0}{H}\, dΣ\end{equation}和等式成立当且仅当两个浸入具有相同的形状算子。在这种情况下，还必须连接$Σ$。在$M_0=\R^{n+1}$的特殊情况下，（\ref{HoloIneq}）中的相等意味着$M$是欧几里得域，并且$ϕ$与$Σ$在$M$中的嵌入全等作为其边界。证明了不等式（\ref{HoloIneq}）蕴含正质量定理（PMT）.注意，使用PMT和$ϕ$是欧氏空间中的严格凸嵌入的附加假设，Shi和Tam \cite{ST1}证明了比（\ref{HoloIneq}）强的积分不等式\begin{equation}\label{shi-tam-Ineq} \int_ΣH\,dΣ\le \int_ΣH_0\, dΣ, \end{equation}。
英文摘要：
Suppose that $Σ=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $Σ$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle\;,\;\rangle_H=H^2\langle\;,\;\rangle$ is at least $n/2$ and equality holds if and only if there exists a parallel spinor field on $ M$. As a consequence, if $Σ$ admits an isometric and isospin immersion $ϕ$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $ M_0$ admitting a parallel spinor field, then \begin{equation} \label{HoloIneq} \int_ΣH\,dΣ\le \int_Σ\frac{H^2_0}{H}\, dΣ\end{equation} and equality holds if and only if both immersions have the same shape operator. In this case, $Σ$ has to be also connected. In the special case where $M_0=\R^{n+1}$, equality in (\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $ϕ$ is congruent to the embedding of $Σ$ in $M$ as its boundary. We also prove that Inequality (\ref{HoloIneq}) implies the Positive Mass Theorem (PMT). Note that, using the PMT and the additional assumption that $ϕ$ is a strictly convex embedding into the Euclidean space, Shi and Tam \cite{ST1} proved the integral inequality \begin{equation}\label{shi-tam-Ineq} \int_ΣH\,dΣ\le \int_ΣH_0\, dΣ, \end{equation} which is stronger than (\ref{HoloIneq}) .
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