Density Estimates for Minimal Surfaces and Surfaces Flowing by Mean Curvature

ROBERT GULLIVER ABSTRACT. Let be a two-dimensional immersed minimal surface in a manifold Mn, having a curve as boundary. We do not assume that has minimum area. It will be shown that the number of sheets of passing through a point p 2 M (the density of at p) will be bounded by geometric measures of the complexity of . However, such an estimate must also depend on the geometry of the ambient manifold M . Suppose that M is simply connected, and that the sectional curvatures of M are less than or equal to a nonpositive constant 2. Let A( ) denote the minimum over p 2 M of the area of the geodesic cone over with vertex p. If for some integer m 0 the total absolute curvature of satisfies Z j~kj ds 2 m+ 2A( ); then the number of sheets through one point is at most m 1. In particular, if this inequality holds with m = 2, then must be embedded. An analogous result holds if M is a hemisphere. We shall also discuss conjectures about analogous estimates for a surface which evolves by its mean curvature vector. The Euclidean case M = Rn was proved by Eckholm, White and Wienholtz [EWW]. This report is based on joint work with Jaigyoung Choe [CG].