Self-Expander of Mean Curvature Flow and Applications

Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it emerges from a cone singularity. The nonuniqueness of self-expanders presents challenges in the study of cone-like singularities in the flow. In this talk, I will discuss some recent development on a variational theory for self-expanders and an application to the question on lower density bounds for minimal cones.