Pseudo-convex submanifolds — curvature properties, Plateau problem, and the discrete theory
As a generalization of the usual convex hypersurfaces, we consider a spacelike submanifold immersed in a pseudo Euclidean space, whose normal bundle is induced with a Lorentz inner product of signature (1,p), and we make a convexity assumption on its second fundamental form. The simplest example is a closed curve in the 3-dimensional Lorentz space whose tangent and normal vectors span a spacelike plane, and it winds around some timelike axis with index 1. For such a pseudo-convex loop, we show that it satisfies a reversed Fenchel type inequality (its total curvature is no more than 2pi), and it always span a spacelike maximal surface. Then we give the general definition of a pseudo-convex submanifold and demonstrate their close relationship with convex hypersurfaces. If time is allowed we will briefly mention what is a pseudo-convex polyhedron and a discretized Positive Mass Conjecture about the Liu-Yau mass. This is a joint work with Dr. Nan Ye, and the final part is an ongoing project with my postdoc Dong Zhang.