caa a22 4500 606194487 CHVBK 20210128105229.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s00030-014-0287-9 doi (NATIONALLICENCE)springer-10.1007/s00030-014-0287-9 Fully nonlinear curvature flow of axially symmetric hypersurfaces [Elektronische Daten] [James McCoy, Fatemah Mofarreh, Valentina-Mira Wheeler] 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 etal., 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. Springer Basel, 2014 Curvature flow nationallicence Parabolic partial differential equation nationallicence Hypersurface nationallicence Initial-boundary value problem nationallicence Neumann boundary condition nationallicence McCoy James Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut Mofarreh Fatemah Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut Wheeler Valentina-Mira Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/2(2015-04-01), 325-343 1021-9722 22:2<325 2015 22 30 https://doi.org/10.1007/s00030-014-0287-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00030-014-0287-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- McCoy James Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut NATIONALLICENCE 700 1- Mofarreh Fatemah Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut NATIONALLICENCE 700 1- Wheeler Valentina-Mira Institute for Mathematics and its Applications, University of Wollongong, Northfields Av, 2522, Wollongong, NSW, Australia aut NATIONALLICENCE 773 0- Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/2(2015-04-01), 325-343 1021-9722 22:2<325 2015 22 30 SWISSBIB 560474075