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

In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken [13]. We will also study expansions of the Brown-York mass and the Hawking mass of geodesic spheres with center at a fixed point p of a three manifold. Some geometric consequences will be derived.

In this paper, we obtain some rigidity theorems on compact manifolds with nonempty boundary. The results may be related to the positivity of some quasi-local mass of Brown–York type. The main argument is to use monotonicity of quantities similar to the Brown–York quasi-local mass in a foliation of quasi-spherical metrics. Together with a hyperbolic version of positivity of a mass quantity, we obtain our main results.

The problem of a uniform current passing through a circular cylinder submerged below an ice sheet is considered. The fluid flow is described by the linearized velocity potential theory, while the ice sheet is modeled through a thin elastic plate floating on the water surface. The Green function due to a source is first derived, which satisfies all the boundary conditions apart from that on the body surface. Through differentiating the Green function with respect to the source position, the multipoles are obtained. This allows the disturbed velocity potential to be constructed in the form of an infinite series with unknown coefficients which are obtained from the boundary condition. The result shows that there is a critical Froude number which depends on the physical properties of the ice sheet. Below this number there will be no flexural waves propagating to infinity and above this number there will be two waves, one on each side of the body. When the depth based Froude number is larger than 1, there will always be a wave at far upstream of the body. This is similar to those noticed in the related problem and is different from that in the free surface problem without ice sheet. Various results are provided, including the properties of the dispersion equation, resistance and lift, ice sheet deflection, and their physical features are discussed.