Y. L. Xin

Minimal submanifolds and related topics

Bernsteins Theorem and Its Generalizations Weierstrass Type Representations Plateaus Problem and Douglas-Rato Solution Minimal Submanifolds of Higher Codimension Stable Minimal Hypersurfaces Bernstein Type Theorems for Higher Codimension Entire Space-Like Submanifolds.

Some Aspects of the global Geometry of Entire Space-Like Submanifolds

We prove some Bernstein type theorems for entire space-like subma-nifolds in pseudo-Euclidean space and as a corollary, we give a new proof of the Calabi-Pogorelov theorem for Monge-Ampère equations.

The rigidity theorems of self-shrinkers

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.

A spherical Bernstein theorem for minimal submanifolds of higher codimension

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptRIV(2)) to any Riemannian manifold with finite energy has to be constant.

The characteristic numbers of 4-dimensional Kähler manifolds

Using the decomposition of the curvature tensor, we obtain relations between the Euler number and the Pontrjagin number for 4-dimensional k-Ricci pinched and λ-holomorphically pinched Kahler manifolds.