We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmannian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.

We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

We consider a 1-parameter family of strictly convex hypersurfaces in Rn+1 moving with speed −Kαν, where ν denotes the outward-pointing unit normal vector and α⩾1/(n+2). For α>1/(n+2), we show that the flow converges to a round sphere after rescaling. In the affine invariant case α=1/(n+2), our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2dimT and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case.

The paper deals with a variational approach of the subRiemannian geometry from the point of view of Hamilton-Jacobi and Hamiltonian formalism. We present a discussion of geodesics from the point of view of both formalisms, and prove that the normal geodesics are locally length-minimizing horizontal curves.