Self-similar solutions to the curve shortening flow

Entire Self-similar Solutions to Lagrangian Mean Curvature Flow

Abstract. We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function u has eigenvalues strictly uniformly between −1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one-to-one correspondence to functions of homogenous of degree 2 with th...

An Introduction to Curve-Shortening and the Ricci Flow

Self-similar expanding solutions for the planar network flow

We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect t...

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...

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.