We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in R n . As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypersurface. Statement of the ResultMonotone quantities along hypersurfaces evolving under the inverse mean flow have many applications in geometry and relativity. In [3], Huisken and Ilmanen applied the monotone increasing property of Hawking mass to give a proof of the Riemannian Penrose Inequality. In a recent paper [1], Brendle, Hung and Wang discovered a monotone decreasing quantity along the inverse mean curvature flow in Anti-Desitter-Schwarzschild manifolds and used it to establish a Minkowski-type inequality for star-shaped hypersurfaces.In this note, we provide a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in R n :Theorem 1. Let Σ be a smooth, closed, embedded hypersurface with positive mean curvature in R n . Let I be an open interval and X : Σ × I → R n be a smooth map satisfying ∂X ∂t = 1 H ν, (1.1)

We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type results and some eigenvalue estimates for hypersurfaces.

In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

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In this paper, we obtain a positivity result of a quasi-local mass integral as proposed by Shi and Tam in general dimensions. The main argument is based on the monotonicity of a mass integral in a foliation of quasi-spherical metrics and a positive mass type theorem which was proved by Wang and Yau in the three-dimensional case, and is shown here in higher dimensions using spinor methods.The well-known positive mass theorem states that for a complete asymptotically flat manifold (M, g) such that g behaves like Euclidean at infinity near each end and suppose its scalar curvature is non-negative, then its ADM mass (cf.[2] for its definition and motivation) of each end is non-negative. Moreover, if the ADM mass of one of the end is zero, then (M, g) is actually the Euclidean space. The positive mass theorem was proved by Schoen and Yau [11,12] using minimal surface techniques. Later on, Witten [17] (see also [4,10]) gave a simplified proof the positive mass theorem using the spinors. Since then the method of spinors has been adopted by many people to prove positive mass-type theorems or some rigidity results; see for example [1,9,13,15,16].In particular, let us look at some results in this direction. Wang and Yau [15] developed a quasi-local mass for a three-dimensional manifold with boundary whose scalar curvature is bounded from below by some negative constant. Using spinors methods, they were able to prove that this mass is non-negative. Later on, Shi and Tam [14] also proved a similar result in the three-dimensional case, but with a simpler and more explicit definition of the mass. In this paper, we will show that the results of Shi-Tam and Wang-Yau also hold in higher dimensions.More