Brett Kotschwar

A local version of Bando's theorem on the real-analyticity of solutions to the Ricci flow

It is a theorem of S. Bando that if g(t) is a solution to the Ricci flow on a compact manifold M, then (M,g(t)) is real-analytic for each t > 0. In this note, we extend his result to smooth solutions on open domains UM.

A short proof of backward uniqueness for some geometric evolution equations

We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman inequalities. We also demonstrate the applicability of the technique to the

An Energy Approach to Uniqueness for Higher-Order Geometric Flows

L^2

Kählerity of Shrinking Gradient Ricci Solitons Asymptotic to Kähler Cones

-curvature flow and other higher-order equations.

Time-analyticity of solutions to the Ricci flow

We describe a simple, direct method to prove the uniqueness of solutions to a broad class of parabolic geometric evolution equations. Our argument, which is based on a prolongation procedure and the consideration of certain natural energy quantities, does not require the solution of any auxiliary parabolic systems. In previous work, we used a variation of this technique to give an alternative proof of the uniqueness of complete solutions to the Ricci flow of uniformly bounded curvature. Here we extend this approach to curvature flows of all orders, including the

An energy approach to the problem of uniqueness for the Ricci flow

L^2

On rotationally invariant shrinking gradient Ricci solitons

L2-curvature flow and a class of quasilinear higher-order flows related to the obstruction tensor. We also detail its application to the fully nonlinear cross-curvature flow.

Backwards uniqueness of the Ricci flow

We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.