This is joint work with Minghao Wang. In this talk, I will explain the convergence of Feynman graph integrals on closed real-analytic K\"ahler manifolds: Using Getzler’s rescaling technique, the graph integrands extend naturally to the Fulton–MacPherson compactification as forms with divisorial singularities, allowing a rigorous definition as Cauchy principal value integrals. As an application, this yields a mathematical construction of the higher-genus B-model invariants on Calabi–Yau threefolds in the sense of Bershadsky–Cecotti–Ooguri–Vafa (BCOV).

Abstract: The classical Yamabe problem—solved through the work of Yamabe, Trudinger, Aubin, and Schoen—asserts that on any closed smooth connected Riemannian manifold $(M^n,g)$, $n\geq 3$, one can find a metric conformal to $g$ with constant scalar curvature. A fully nonlinear analogue replaces the scalar curvature by symmetric functions of the Schouten tensor. Traditionally, the existing theory has required the scalar curvature to have a fixed sign. In a recent work, we broaden the scope of fully nonlinear Yamabe problem by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results. Our results allow conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities. I will also discuss extensions to manifolds with boundary, treating prescribed boundary mean curvature and the boundary curvature arising from the Chern–Gauss–Bonnet formula.

Joint with Analysis seminar at 3pm.

Weighted-cscK metrics provide a universal framework for the study of canonical metrics, e.g., extremal metrics, Kahler-Ricci soliton metrics, \mu-cscK metrics. In joint works with Yaxiong Liu, we prove that on a Kahler manifold X, the G-coercivity of weighted Mabuchi functional implies the existence of weighted-cscK metrics.  In particular, there exists a weighted-cscK metric if X is a projective manifold that is weighted K-stable for models.  We will also discuss some progress on singular varieties.