Fully nonlinear curvature flow of axially symmetric hypersurfaces
- 127 Downloads
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.
Mathematics Subject ClassificationPrimary 35K55 35R35 53C44 Secondary 35K60
KeywordsCurvature flow Parabolic partial differential equation Hypersurface Initial-boundary value problem Neumann boundary condition
- 6.Andrews, B.H., Langford, M., McCoy, J.A.: Convexity estimates for fully non-linear surface flows. J. Differ. Geom. (to appear)Google Scholar
- 18.McCoy, J.A., Mofarreh, F.Y.Y., Williams, G.H.: Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions. Annali di Matematica Pura ed Applicata, 1–13 (2013, to appear in print). doi: 10.1007/s10231-013-0337-7