Fully nonlinear curvature flow of axially symmetric hypersurfaces

Abstract

Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds ( J. A. McCoy et al., Annali di Matematica Pura ed Applicata 1-13, 2013). In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (B. H. Andrews, Invent Math 138(1):151-161, 1999; Calc Var Partial Differ Equ 39(3-4):649-657, 2010); these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3. Abstract. Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions was considered and a partial characterisation of the finite maximal time of existence was obtained, in the case of convex speeds [MMW]. In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces [An2, An4]; these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3.Mathematics Subject Classification (2010). Primary 35K55, 35R35, 53C44; Secondary 35K60.

More Mixed Volume Preserving Curvature Flows

“…where again we have used that M t 0 is convex so H > 0. It follows that the maximum of G does not increase and therefore, as in [30], the pinching ratio does not deteriorate under the flow (1). Since M 0 was strictly convex, that the pinching ratio does not deteriorate implies that M t remains strictly convex, as long as the solution to (1) exists.…”

“…For α ≥ 1 this term is clearly nonpositive. Now, as in [30], for the gradient terms to have the right sign requires the pinching ratio to be not greater than 1 + 2 α−1 . The same argument as in the proof of Proposition 4.2 then shows that the pinching ratio does not deteriorate in the case of evolving axially symmetric hypersurfaces.…”

“…using Lemma 2.1 (i) and our assumption that M t 0 is convex. Moreover, as in [30], the gradient term in Lemma 4.1 is, in view of Lemma 2.3 equal to 4 n (n − 1) f…”

“…In view of the result of [33] and since the relevant results of [9,30] continue to hold where the degree of homogeneity of the speed is α > 1 it is natural to consider whether the corresponding constrained flows with F homogeneous of degree α > 1 also evolve suitable initial hypersurfaces to spheres. In this section we show that this is indeed the case.…”

Variational discretization of axisymmetric curvature flows

“…[8, (1.14)], where in (1.7) we again assume that n S is the outer normal to Ω(t) on S(t). Such flows have found considerable interest in geometry recently and we refer to [26,40,39,42] for more information. One reason why the Gauss curvature flow is of particular interest, is because this flow allows to study the fate of the rolling stones, see [2].…”