Evolving convex hypersurfaces

Abstract

The subject of this thesis is the deformation of hypersurfaces by means of geometrically defined parabolic equations. In the most general case, we consider hypersurfaces moving in a Riemannian manifold, with speed determined by a function of the Weingarten curvature. The majority of the thesis concerns the case of convex hypersurfaces. Section I of the thesis concerns evolution equations which generalise, in a certain sense, the well-known mean curvature flow : We consider the motion of hypersurfaces in Euclidean space, where the speed is a function of the principal curvatures. As for the mean curvature flow, we require this function to be homo-geneous of degree one, and strictly increasing in each argument.The motion is then described by a fully nonlinear parabolic equation. Under natural structure conditions on the equations and natural convexity conditions on the initial hyper- surface,it is shown that a unique solution exists for a finite time;this solution converges uniformly to a point and becomes spherical in shape towards the final time.This result generalises work on the mean curvature flow by Gerhard Huisken, and related work on other particular flows by Ben Chow.The proof employed is in some respects similar to these earlier cases,but achieves important simplifications through the use of a new result concerning locally pinched convex hypersurfaces. In section II, we consider a much wider class of parabolic evolution equa- tions, allowing not only other degrees of homogeneity for the speed, but also non- homogeneous equations, and speeds depending on the normal direction at each point as well as the Weingarten curvature. Precise Harnack estimates are proved for a very wide class of such equations, characterised by simple structure condi- tions. In the proof, the Gauss map is used to parametrise the hypersurfaces. This change in parametrisation results in remarkable simplification and clarification of the calculations. In contrast, long and complicated calculations were required by Hamilton and Chow in their proofs of special cases of the Harnack inequal- ities. Some entropy inequalities are also proved here for special flows,and the calculation of the Harnack inequalities is extended to the case of complete convex hypersurfaces. Section III gives results for a wide variety of flows: The first chapter deals with a class of contraction flows, showing under appropriate conditions that convex hypersurfaces contract to points. The next chapter concentrates on contracting curves (a case not considered in section 1). A natural class of anisotropic flows is considered,allowing homogeneity of degree greater than or equal to one in the curvature. It is shown that embedded convex curves have the expected limiting behaviour under such equations.The results are proved using generalisations of methods due to Gage. The third chapter concerns anisotropic expansion flows, showing under appropriate conditions that star-shaped hypersurfaces expand to infinite radius, converging to the expected limiting shape as they do so. This generalises results for the isotropic case due to Gerhardt and Urbas. Section IV uses the Gauss map techniques to give an elegant new proof of the Aleksandrov-Fenchel inequalities for mixed volumes of convex regions. The proof uses special evolution equations whose form is suggested by expressions for the mixed volumes.The proof is significantly simpler than those previously available. In section V it is shown that there is an important connection between entropy inequalities and the Aleksandrov-Fenchel inequalities.New entropy inequalities are proved for many evolution equations, by a new proof which directly uses the Aleksandrov-Fenchel inequalities.These estimates are applied to expansion flows of curves in the plane,with speeds homogeneous of degree less than minus one in the curvature.It is shown that solutions expand to infinity in finite time,and that they approach spheres (or other limit shapes for anisotropic equations) after rescaling.Another interesting consequence is that solutions to contraction flows with small degree of homogeneity do not in general converge to the expected limiting shape near the final singularity. The last section concerns hypersurfaces in Riemannian background spaces. The techniques of section I are adapted to this more difficult situation, giving good results for a somewhat restricted class of flows. A strictly convex, compact initial hypersurface,in a space with non-negative sectional curvatures, gives a solution which contracts to a point and becomes round. Also, slightly different flows are used to give the same result for an initial hypersurface with all principal curvatures greater than 1,in a background space with all sectional curvatures greater than or equal to -1. It follows that any such hypersurface is the boundary of an immersed disc.This gives an elegant new proof of the 1/4-pinching sphere theorem of Klingenberg, Berger and Rauch, and also proves a generalised "dented sphere" theorem which allows some negative curvature.