Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature

Abstract

Abstract. In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem. §0 Introduction. The structure of a manifold with positive or nonnegative scalar curvature has been studied extensively. There are many beautiful results for compact manifolds without boundary, see [L,. For example, in [L], Lichnerowicz found that some compact manifolds admit no Riemannian metric with positive scalar curvature. In [SY1-2] Schoen and Yau proved that every torus T n with n ≤ 7 admits no metric with positive scalar curvature, and admits no non-flat metric with nonnegative scalar curvature. This is also proved later by Gromov and Lawson [GW3] for n > 7.For complete noncompact manifolds, the most famous result is the Positive Mass Theorem (PMT), first proved by Schoen and Yau [SY3-4] and later by Witten [Wi] using spinors, see also [PT,B1]. One of their results is as follow: Suppose (M, g) is an asymptotically flat manifold such that g behaves like Euclidean at infinity near each end, and suppose its scalar curvature is nonnegative, then (M, g) is actually flat if the ADM mass of one of the ends is zero.It is natural to ask what we can say about manifolds with boundary and with nonnegative scalar curvature. In a recent work of Yau [Y], it was proved that if Ω is a noncompact complete three manifold with boundary and with scalar curvature not less than −3/2c 2 . Suppose one of the component of ∂Ω has nonpositive Euler number and mean curvature

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