We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply that the equality holds only for smooth points.

In this paper, we prove the existence of a solution for the exterior Dirichlet problem for Hessian equations on a non-convex ring. Moreover, the solution we obtained is smooth. This extends the result of [Bao-Li-Li, “On the exterior Dirichlet problem for Hessian equations” Trans. Amer. Math. Soc. 366(2014)].

We prove that the tangent cone at the first blow-up time of the mean curvature flow of a closed symplectic surface in a compact Kahler-Einstein surface consists of a finite union of planes in R⁴. Furthermore, when the flow develops a Type I^* singularity at (X₀,T), then the tangent cone is a holomorphic cone.

We establish global existence of the pluriclosed flow with arbitrary initial data on Oeljeklaus-Toma manifolds, and Gromov-Hausdorff convergence of blowdown limits to a torus under natural conjectural bounds on the flow at infinity. In the case of generalized Kahler-Ricci flow we prove refined a priori estimates in support of these conjectural bounds.

We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^{d}$ depending on the first and second fundamental forms and their covariant derivatives.