Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary

Details Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary

2003-05-19

arxiv.org/abs/math/0305263v1

Mathematics

Differential Geometry

12 pages, latex2e

Scientific paper

In [5] Herzlich proved a new positive mass theorem for Riemannian 3-manifolds $(N, g)$ whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass $m(g)$ must be strictly positive). We prove that the mass is bounded from below by a positive constant $c(g), m(g) \geq c(g)$, and the equality $m(g) = c(g)$ holds only if, outside a compact set, $(N, g)$ is conformally flat and the scalar curvature vanishes. The constant $c(g)$ is uniquely determined by the metric $g$ via a Dirac-harmonic spinor.

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