In this paper, we solve a Monge-Ampre-type equation by the continuity method. This equation is motivated by the superstring theory.

The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold M, based on a geodesic metric. We present a formal definition of the skeleton S(\Omega) for a shape V in M and show several properties that make S(\Omega) distinct from its Euclidean counterpart in R^2. We further prove that for a shape sequence {\Omega_i} that converge to a shape V in M, the mapping \Omega\rightarrow S(\Omega_i) is lower semi-continuous. A direct application of this result is that we can use a set P of sample points to approximate the boundary of a 2D shape V in M, and the Voronoi diagram of P inside \Omega\subset M gives a good approximation to the skeleton S(\Omega). Examples of skeleton computation in topography and brain morphometry are illustrated.

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Given a finite set of r points in a closed surface of genus r , we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if r for some integer r .

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem which applies only to calibrated submanifolds of special holonomy ambient manifolds.