Time: 10:00-10:50, Thursday, September 3 2026
Venue: E14-301
Speaker：Wenkui Du, Hunan University
Title：Regularity of Limiting Interfaces Arising from Allen Cahn Equation on Riemannian Manifolds with Boundary
University：In this talk, I will discuss the singularity structure of geodesic networks on two dimensional surfaces with boundary and the regularity of free boundary minimal hypersurfaces arising in the min-max theory of geodesics and minimal surfaces. Our approach is based on the study of regularity estimates of interfaces of the ε-phase transition Allen–Cahn equation on manifolds with boundary, extending the pioneering interior works of Wang–Wei and Chodosh– Mantoulidis to the boundary setting. In particular, I will describe curvature estimates for interfaces of stable solutions of Allen–Cahn equation on manifolds with boundary. Then I will discuss boundary regularity of limiting interfaces of solutions of Allen-Cahn equation with uniformly bounded energy and Morse index. As the first application, using the above curvature estimates and the remarkable work of Liu-Wei, we show the billiard regularity conjecture and boundary version p-width representation conjecture of Chodosh–Cholsaipant. As the second application, we establish the regularity of free boundary minimal hypersurfaces in three and four dimensional manifolds. This work is based on my collaboration with Akashdeep Dey, Lorenzo Sarnataro and Davide Parise.
