We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces$S$which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound$b$. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0.

In this paper, we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding toolthe hyperbolic geometric flow. This kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds. It possesses many interesting properties from both mathematics and physics. Several applications of this method have been found.

We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K¨ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the 3-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.