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.

Abstract: The geometric flow of hypersurfaces is an interesting and active area. Its importance lies in the applications in geometry and topology. For example, Huisken and Ilmanen in 2001 applied the inverse mean curvature flow to prove the famous Penrose conjecture. Brendle-Guan-Li proposed a conjecture on the Alexandrov-Fenchel inequalities for hypersurfaces in the sphere and introduced a locally constrained fully nonlinear curvature flow to study this conjecture.  In this talk, we will discuss using a new type of flow to study this question and some recent progress on this conjecture.