Quadratic Flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Explore advanced mathematical concepts in geometric measure theory through this research lecture examining quadratic flatness and regularity properties for codimension-one varifolds with bounded anisotropic mean curvature. Delve into recent collaborative work with Mario Santilli that demonstrates the existence of open and dense subsets where varifolds can be touched by mutually tangent balls, leading to quadratic height decay and regularity results. Learn about the application of Allard's 1986 regularity theorem to establish that certain points are regular points of class (1,α) for any α between 0 and 1. Discover how the research proves that regular regions are almost equal to subsets where blow-up limits are not the whole space, a condition that appears weak but offers hope for broader regularity results. Gain insight into the historical development of regularity problems for varifolds satisfying bounds on anisotropic first variation, and understand the main proof techniques used to establish these significant theoretical results in geometric analysis.