Consider a compact K¨ahler manifold Mm with Ricci curvature lower bound RicM ≥ −2 (m + 1) . Assume that its universal cover fM has maximal bottom of spectrum 1 fM = m2. Then we prove that fM is isometric to the complex hyperbolic space CHm.

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

We consider inverse curvature ows in Hn+1 with star-shaped initial hypersurfaces and prove that the ows exist for all time, and that the leaves converge to innity, become strongly convex exponentially fast and also more and more totally umbilic. Afteran appropriate rescaling the leaves converge in C1 to a sphere.

We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.