Damin Wu

The equality case of the Penrose inequality for asymptotically flat graphs

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.

Negative holomorphic curvature and positive canonical bundle

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

Semilinear equations, the

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the

\gamma_k

\gamma_k

function, and generalized Gauduchon metrics

functions on the space of its hermitian metrics.

Picard number, holomorphic sectional curvature, and ampleness

We prove that for a projective manifold with Picard number equal to one, if the manifold admits a Kähler metric whose holomorphic sectional curvature is quasi-negative, then the canonical bundle of the manifold is ample.