Geometric Analysis Seminar: Yusen Xia (UCSB)

Title: Log-concavity and the geometry of eigenfunctions on ℂP²

Abstract: When does the first Dirichlet eigenfunction inherit the convexity of its domain? The Euclidean log-concavity theorem of Brascamp and Lieb, and its role in fundamental-gap estimates, suggest a close connection between domain geometry and the shape of the ground state. On complex projective space, however, the curvature distinguishes complex directions from transverse ones, obstructing a direct extension of the usual Hessian maximum-principle argument. 

I will present quantitative log-concavity estimates for the first Dirichlet eigenfunction on smooth connected 1-convex domains in ℂP². The proof combines a gradient-dependent barrier with a first-contact argument. I will discuss how these estimates yield uniform convexity of regular level sets and a lower bound for the Dirichlet fundamental gap