Title: Analysis of weighted p-harmonic forms and applications
Speaker: Nguyen Thac Dung (Hanoi University of Science)
Abstract:
In this talk, we investigate weighted p-harmonic forms on smooth metric measure space with a weighted Sobolev or a weighted Poincare inequality. As applications, we conclude Liouville property for weighted p-harmonic functions/forms/maps and study p-hyperbolic ends.
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Title: Steklov Eigenvalues and Free Boundary Minimal Surfaces
Speaker: Tran Thanh Hung (Texas Tech.)
Abstract:
The connection between Steklov eigenvalues and free boundary minimal surfaces has been developed to great effects in the last couple of years. In this talk, we first describe that connection between PDE and geometry. Then, we'll introduce the notion of a Jacobi-Steklov eigenvalue and see how it reveals extremal properties of a free boundary minimal surface.
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Title: Connectedness and embeddedness of minimal submanifolds
Speaker: Keomkyo Seo (Sookmyung Women's University)
Abstract:
It is well-known that any simple closed curve in $\mathbb{R}^3$ bounds at least one minimal disk, which was independently proved by Douglas and Rad\'{o}. However, for any given two disjoint simple closed curves, we cannot guarantee existence of a compact connected minimal surface spanning such boundary curves in general. From this point of view, it is interesting to give a quantitative description for necessary conditions on the boundary of compact connected minimal surfaces. We derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small $L^2$ norm of the mean curvature vector in a Riemannian manifold. Moreover, it follows from the density estimates that certain compact surfaces with small $L^2$-norm of the mean curvature vector in a Riemannian manifold must be embedded.