NDLI: Stability of Near Cylindrical Stationary Solutions to Weighted-Volume Preserving Curvature Flows

Content Provider	Springer Nature Link
Author	Hartley, David
Copyright Year	2015
Abstract	In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In previous work the stability of cylinders under the flows was considered; it was found that they are stable provided the radius satisfies a certain condition. Here we consider flows where there is a critical radius such that the cylinders are only stable on one side of the critical value, and the weight function is a linear combination of the elementary symmetric functions, the reason for such a choice is made clear in the appendix. We find that in such instances bifurcation from the cylindrical stationary solutions occurs at the critical radius and we determine a condition on the speed and weight functions such that the nearby non-cylindrical stationary solutions are stable under the flow. In particular, we find that high dimensional half-period unduoids close to the cylinder are stable stationary solutions to the volume preserving mean curvature flow under axially symmetric, volume preserving perturbations. We will also highlight the specific cases of homogeneous speed functions and the mixed-volume preserving mean curvature flows.
Starting Page	2169
Ending Page	2203
Page Count	35
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	26
Issue Number	3
e-ISSN	1559002X
Language	English
Publisher	Springer US
Publisher Date	2015-06-27
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Curvature flows Constant mean curvature hypersurfaces Stability Geometric evolution equations Nonlinear parabolic equations Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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