Gradient and oscillation estimates and their applications in geometric PDE

We describe some recent ‘oscillation’ estimates in geometric PDE, where estimates are produced using the maximum principle applied to functions depending on several points. Applications include sharp short-time regularity results, sharp long-time behaviour which related closely to optimal estimates on eigenvalues, and elegant proofs of several key results on geometric evolution equations. 1. A brief history of gradient estimates Gradient estimates are bread and butter in PDE theory, but I want to concentrate here on a specific part of that wide picture. The techniques I have in mind are gradient estimates proved using the maximum principle in the spirit of the work of Cheng and Yau [CY] on eigenfunctions and solutions of semilinear equations, and Yau’s gradient estimates for harmonic functions [Y1], which in particular gave non-existence of non-constant bounded harmonic functions on complete manifolds of non-negative Ricci curvature. More specifically the topics I will discuss later will have connections with work which used such gradient bounds to prove the first eigenvaue of the Laplacian on a compact manifold, beginning with the work of Peter Li [Li] who gave a lower bound on λ1 in terms of diameter for manifolds with non-negative Ricci curvature. This was developed by Li and Yau [LY] to a sharper estimate (which I will described briefly below), and culminated in the work of Zhong and Yang [ZY] who gave a sharp lower bound on λ1 in terms of diameter for manifolds with non-negative Ricci curvature. There is a more general picture which I will discuss later. Before going on to the main topic (which is not gradient estimates but a somewhat different technique) I want to briefly outline how gradient estimates yield eigenvalue estimates in the work of Li and Yau. Suppose we have an eigenfunction u on a compact manifold M of non-negative Ricci curvature, so that ∆u+ λu = 0 everywhere. The eigenfunction can be normalized by scaling so that it takes values in the range [−a, 1] for some a ∈ (0, 1]. Li and Yau proved the following gradient 2010 Mathematics Subject Classification. Primary 35K10, 35K55, 35P15, 58J60. c ©2011 American Mathematical Society and International Press