Classification of compact ancient solutions to the curve shortening flow

An Introduction to Curve-Shortening and the Ricci Flow

The Blow up Analysis of Solutions of the General Curve Shortening Flow

In this paper, a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time under the general curve shortening flow.

Curve Shortening Flow in a Riemannian Manifold

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let M be a compact locally symmetric space. If the curve shortening flow exists for...

Ancient Solutions to Kähler-ricci Flow

In this paper, we prove that any non-flat ancient solution to KählerRicci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results then are applied to study the geometry ...

Ancient Solutions of the Affine Normal Flow

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques...