Local and global analysis of geometric partial differential equations and their application to curvature flow problems

Abstract

“An analytical approach to many problems in geometry leads to the study of partial differential equations.” (A. V. Pogorelov, Foreword to The Minkowski Multidimensional Problem) This thesis concerns itself with three problems lying in the intersection between partial differential equations (PDEs) and geometric analysis. A modified version of the mean curvature flow of convex surfaces whose flow speed is a nonhomogeneous function of the principal curvatures is studied in Chapter 2. After proving short-time existence, we show using methods developed by Chow in 1985 that if the principal curvatures of the initial hypersurface satisfy a certain pinching condition, then this is preserved by the flow. We then apply the pinching estimate to prove that the flow converges to a sphere under rescaling. In Chapter 3, we investigate the existence problem for an S4-rotationally-symmetric, compact self-similar shrinking solution (self-shrinker) of the mean curvature flow in R 3 constructed numerically by Chopp in 1994. We provide an interior gradient estimate for the self-shrinker PDE, and also explore the problem from a new geometrical viewpoint, enabling us to transform the PDE into a Monge-Amp`ere-type equation. Besides this, we demonstrate that standard methods for second order quasilinear PDEs, such as the method of continuity, fail to prove existence in this context. The climax of the thesis is Chapter 4: a thorough examination of Monge-Amp`ere-type equations on annular domains satisfying mixed boundary conditions, with a view towards Gauss curvature flow and the equation of prescribed Gauss curvature. We focus on domains Ω whose boundary consists of two smooth, closed, convex hypersurfaces containing the origin, impose a homogeneous Dirichlet condition on the outer boundary and a Neumann condition on the inner boundary, and consider smooth, convex solutions of the Monge-Amp`ere-type equation det[D2u] = ψ n(x, u, Du). Original results within this chapter include a priori estimates under certain structure conditions, existence results under said conditions, and examples of the colourful behaviour exhibited by solutions of these highly nonlinear equations. In particular, we construct counterexamples demonstrating the necessity of extra restrictions on the Neumann condition and the principal curvatures of the inner boundary in order to obtain global C 2 estimates. We show that under these conditions, the problem admits a smooth solution. However, we also show that, for some choices of ψ, even global C 1 estimates cannot be proven. In these cases, a local estimate for |Du| near the inner boundary is derived.