Geometric Analysis and Topology Seminar

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A Local Higher Regularity Theorem of the HCMA Equation

Speaker: Qi Yao, Stony Brook University

Location: Warren Weaver Hall 101

Date: Friday, February 20, 2026, 10 a.m.

Synopsis:

The Homogeneous Complex Monge-Ampère (HCMA) equation plays a central role in Kähler geometry, effectively describing geodesics in the space of Kähler metrics. A major open question concerns the regularity of weak solutions to this equation.

In this talk, I will present a new local higher regularity result for the HCMA equation on complete Kähler manifolds. The proof relies on a local foliation of the space by holomorphic discs and redevelopment of the global pluripotential framework.I will explain a subtle regularity issue in the parameter dependence of these foliations, and how it can be resolved using BMO estimates and a Nash–Moser iteration. By constructing a global psh subsolution, I will show that the local solution determined by the foliation agrees with the global C^{1,1} solution. As an application, I will discuss the consequences of this regularity on the ALE end. 

Notes:

Please note unusual time and location