Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations

Abstract

Estimates on moduli of continuity, isoperimetric profiles of various kinds, and quantities involving function values at several points have been central in several recent results in geometric analysis. This paper surveys some of the techniques and their applications. 1. INTRODUCTORY COMMENTS In this article I want to describe some techniques which have been applied with some success recently to a variety of problems, ranging from my proof with Julie Clutterbuck of the sharp lower bound on the fundamental gap for Schrodinger operators [8] to Brendle’s proof of the Lawson conjecture [29] and my proof with Haizhong Li of the Pinkall-Sterling conjecture [17]. I will discuss several other interesting applications below. The common thread in these techniques is the application of the maximum principle to functions involving several points or to functions depending on the global structure of solutions. Such techniques are not entirely new: In the analysis of parabolic equations in one variable, functions involving two points have been used extensively, going back to the work of Kružkov [61]. Later work using related ideas include the ‘concavity maximum principle’ of Korevaar [59], who considered the function C(x, y, μ) = u(μy + (1− μ)x)− μu(y)− (1− μ)u(x) for μ ∈ (0, 1) and x, y in the domain of u, with u satisfying some elliptic or parabolic equation. He used a maximum principle to prove non-positivity of C, implying convexity of the function u, for several situations of interest. Korevaar’s argument was later extended by several authors, notably Kawohl [55] and Kennington [56, 57]. I will mention later some contributions of Hamilton [51, 52] and Huisken [54] where related ideas were developed for applications in geometric flow problems. It should be pointed out also that very similar ideas arise in the theory of viscosity solutions as expounded by Crandall, Ishii and Lions [41] — for example the proof of the comparison principle there is based on the use of a two-point function to characterise semi-jets of a non-smooth function, and the basic computations involved are similar to some of those I will employ below. 2. PRESERVING MODULUS OF CONTINUITY IN HEAT EQUATIONS Perhaps the simplest example of applying the maximum principle to a function of several points arises in controlling the modulus of continuity of a solution of a parabolic equation. Consider a smooth solution of the heat equation in Rn, either on a domain with suitable boundary conditions (Neumann is simplest but others can also be handled) or otherwise with good control — for simplicity let us consider here only the case where the solution is spatially periodic in some lattice. We will consider mostly the heat equation, but the techniques work well also for a wider class of parabolic equations, which might be described as ‘quasilinear equations with gradient-dependent coefficients’. In particular these are quasilinear flows which are invariant under rigid motions of space, and vertical translation of the graph. In the Euclidean setting these equations preserve any initial modulus of continuity: More generally, for any fixed vector w, any inequality of the form u(x + w) − u(x) ≤ f(w) is Date: November 21, 2014. This survey describes work supported by Discovery Projects grants DP0985802, DP120102462 and DP120100097 of the Australian Research Council. 1