Nonlinear elliptic and parabolic equations of the second order

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of $$\mathbb {Q}$$
 -Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman’s $$\lambda $$
 -entropy functional on K-unstable Fano manifolds are also given.

K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functional along weak geodesic rays in the space of all bounded positively curved metrics on the anti-canonical line bundle of X and also gives a new proof in the non-singular case. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Furthermore, applications to geodesic stability, bounds on the Ricci potential and Perelman's entropy functional on K-unstable Fano manifolds are given.

A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.

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Geometric flows with rough initial data

In this paper, we study the Calabi flow on a polarized Kahler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c^3,a(M) initial Kahler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kahler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kahler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.

On the Calabi Flow

We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a compactness theorem in space of the kahler metrics given unifrom Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-sigularity result about a weak constant scalar curveture metric in a punctured ball. One of the main difficulties is that the metric is not assumed to be smooth outside of the singular point.

On the Calabi flow

https://people.cas.uab.edu/~simanyi/weiyong_he.pdf

Frankel conjecture and Sasaki geometry

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $${{\mathbb R}^{2n}}$$
 , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t = 0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L
 ∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.

Lagrangian mean curvature flow for entire Lipschitz graphs

We shall consider the regularity of solutions for the complex Monge-Ampère equations in Cn or a bounded domain. First we prove interior C2 estimates of solutions in a bounded domain for the complex Monge-Ampère equations with the assumption of an Lp bound for u, p > n2, and of a Lipschitz condition on the right-hand side. Then we shall construct a family of Pogorelov-type examples for the complex Monge-Ampère equations. These examples give generalized entire solutions (as well as viscosity solutions) of the complex Monge-Ampère equation det(uij̄) = 1 in C n.

/pdf/on-the-regularity-of-the-complex-monge-ampere-equations-12amgav0r0.pdf

Weiyong He

Papers

On the Calabi Flow

On the Calabi flow

Frankel conjecture and Sasaki geometry

Lagrangian mean curvature flow for entire Lipschitz graphs

On the regularity of the complex Monge-Ampère equations