Abstract:
In this paper, we study the boundary behavior of the negatively curved K\"ahler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the potential of the KE metric near the boundary $D$. In the more general singular case ($D$ being assumed effective though), we show that the KE metric has mixed cone and cusp singularities near $D$ on the snc locus of the pair. As a corollary, we derive the behavior in codimension one of the KE metric of a stable variety.

Abstract:
We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

Abstract:
In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $\gamma_k$ functions on the space of its hermitian metrics.

Abstract:
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on K\"ahler manifolds of nonnegative orthogonal bisectional curvature.

Abstract:
We construct a compact K\"ahler manifold of nonnegative quadratic bisectional curvature, which does not admit any K\"ahler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional K\"ahler C-space with second Betti number equal to 1, and its canonical metric is a K\"ahler-Einstein metric of positive scalar curvature

Abstract:
We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed earlier. This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation.

Abstract:
Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential form level generalization of the classical Calabi--Yau equation. We establish the existence and uniqueness of the equation on complex tori, and prove certain uniqueness and openness on a general K\"ahler manifold.

Abstract:
In this note we show that if a projective manifold admits a K\"ahler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Abstract:
We prove rigidity for hypersurfaces in the unit (n+1)-sphere whose scalar curvature is bounded below by n(n-1), without imposing constant scalar curvature nor constant mean curvature. The lower bound n(n-1) is critical in the sense that some natural differential operators associated to the scalar curvature may be fully degenerate at geodesic points and cease to be elliptic. We overcome the difficulty by developing an approach to investigate the geometry of level sets of a height function, via new geometric inequalities.

Abstract:
We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete non-compact hypersurfaces of finitely many regular ends with nonnegative scalar curvature are weakly mean convex, and prove a positive mass theorem for such hypersurfaces.