Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bounds and Static Quasi-Local Mass of Tori
arxiv(2024)
摘要
A classic result of Shi and Tam states that a 2-sphere of positive Gauss and
mean curvature bounding a compact 3-manifold with nonnegative scalar curvature,
must have total mean curvature not greater than that of the isometric embedding
into Euclidean 3-space, with equality only for domains in this reference
manifold. We generalize this result to 2-tori of Guass curvature greater than
-1, which bound a compact 3-manifold having scalar curvature not less than
-6 and at least one other boundary component satisfying a 'trapping
condition'. The conclusion is that the total weighted mean curvature is not
greater than that of an isometric embedding into the Kottler manifold, with
equality only for domains in this space. Examples are given to show that the
assumption of a secondary boundary component cannot be removed. The result
gives a positive mass theorem for the static Brown-York mass of tori, in
analogy to the Shi-Tam positivity of the standard Brown-York mass, and
represents the first such quasi-local mass positivity result for non-spherical
surfaces. Furthermore, we prove a Penrose-type inequality in this setting.
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