Integrating Factors for Dirac-Schrodinger Operators: Improving Eigenvalue Estimates and Applications to Charged Positive Mass Theorems Outside Horizon(s)

Abstract

Let $(M^n,g)$ denote a Riemannian spin manifold of dimension $n$ with Dirac operator $D$ induced from the Levi-Cevita connection acing on the spinor bundle, $S$ ($D$ is also called the Atiyah-Singer Operator). Let $c:Cl(T{M}^n)\to End(S)$ be the standard representation of the Clifford Algebra as endomorphisms of the spinor bundle. Let $B\in End(S)$ be a zeroth-order endomorphism of the spinor bundle; given an in an orthonormal frame, $e_j\in T{M}^n$ by the expression $B=f^{\alpha }c(e_{\alpha })$ where the sum is taken over multi-indices, $\alpha =(i_{j_m}),m=1,2,3...,k$, $j_1<j_2<...<j_k$ and each $f^{\alpha }\in C^{\infty }(M^n)$. The purpose of this paper is investigate when the Dirac-Schrodinger operator $D+B$ has an integrating factor, i.e. when does there exist an invertible endomorphism $A\in End(S)$ such that $D(AB)=AD+AB$. This has applications to improving eigenvalue estimates for Dirac-Schrodinger operators and proving positive charged positive mass theorems where such operators appear on the boundary. Of particular interest is the case $n=2$, for boundary Dirac operators of this form appear in charged positive mass theorems based on the initial data formulation in mathematical general relativity. It allows us to generalize a theorem of M. Herzlich (set-forth in his attempt to prove the Riemannian Penrose-inequality using spinors, cf. [1]) to a manifold of dimension $n\ge 3$ containing an electric field and symmetric two-tensor representing the second-fundamental form.

References in corpus (1)

• Eigenvalue estimates for the Dirac-Schrödinger operators