We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.

Abstract. We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

ABSTRACT. We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter. INTRODUCTORY COMMENTSIn our previous papers [1, 2] we proved sharp bounds on the modulus of continuity of solutions of various parabolic boundary value problems on domains in Euclidean space. In this paper, our aim is to extend these estimates to parabolic equations on manifolds. Precisely, let (M, g) be a compact Riemannian manifold with induced distance function d, diameter, while ∇ (a(ω)(µ, ν)) = 0 whenever ∇ω = ∇µ = ∇ν = 0). Then we consider solutions to the parabolic equationOur assumptions imply that the coefficients a ij have the formfor some smooth positive functions α and β. Of particular interest are the cases of the heat equation (with α = β = 1) and the p-laplacian heat flows (with α = (p − 1)|Du| p−2 and β = |Du| p−2 ). Here we are principally concerned with the case of manifolds without boundary, but can also allow M to have a nontrivial convex boundary (in which case we impose Neumann boundary conditions D ν u = 0). Our main aim is to provide the following estimates on the modulus of continuity of solutions in terms of the initial oscillation, elapsed time, κ and D:Theorem 1 |u(x, t) − u(y, t)| ≤ 2ϕd(x, y) 2 , t .

Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.