Bennett Chow

Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball

We consider the (degenerate) parabolic equationut=G(▽▽u + ug, t) on then-sphereSn. This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦▽u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationut=G(Δu +cu, ¦x¦,t) on then-ballBn, wherec ≤ λ2(Bn).

A necessary and sufficient condition for Ricci shrinkers to have positive AVR

In this short note we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.

The linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation

We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelmans and Cao--Hamiltons Harnacks on a steady soliton.