Type I Singularities in the Curve Shortening Flow Associated to a Density

On Type-I singularities in Ricci flow

We define several notions of singular set for Type-I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber [15]. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for...

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...

An Introduction to Curve-Shortening and the Ricci Flow

The Singularities of the 3-secant Curve Associated to a Space Curve

Let C be a curve in P3 over an algebraically closed field of characteristic zero. We assume that C is nonsingular and contains no plane component except possibly an irreducible conic. In [GP] one defines closed r-secant varieties to C, r £ N. These varieties are embedded in G, the Grassmannian of lines in P3. Denote by T the 3-secant variety (curve), and assume that the set of 4-secants is fini...

Ergodicity of Stochastic Curve Shortening Flow in the Plane

We study models of the motion by mean curvature of an (1+1) dimensional interface with random forcing. For the well-posedness of the models we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov-Rozovskĭı, replacing the standard coercivity assumption by a Lyapunov type condition. Ergodicity is established for the ...