Mean curvature flow evolves isometrically immersed base manifolds M in the direction of their mean curvatures in an ambient manifold M . If the base manifold M is compact, the short-time existence and uniqueness of the mean curvature flow are well known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.

Abstract. Let M n be a complete noncompact Kähler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by O d (M n ) the space of holomorphic functions of polynomial growth of degree at most d on M n . In this paper we prove thatfor all d > 0, with equality for some positive integer d if and only if M n is holomorphically isometric to C n . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in R 3 . We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into the Lorentz-Minkowski space R 2,1 .

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we extend a result by Colding that an n−dimensional manifold with Ricci curvature greater or equal to n − 1 and volume close to that of the unit n−sphere is close (in the Gromov-Hausdorff distance) to the sphere, from the case of Riemannian manifolds to the case of Alexandrov spaces, with an additional assumption, roughly speaking, that the rough volume of the set of "short" cut points is small, following the basic idea in the Riemannian case with necessary modifications because of the only almost everywhere second differentiability of distance functions.