Paper · Dec 14, 2024
Isotropy invariant graphical mean curvature flows in warped products
As on arXiv
Abstract
In this paper, we study the graphical mean curvature flow in a warped product , where is a symmetric space of compact type, is an open interval, and is a smooth positive function on . If the initial hypersurface is -equivariant, then the -equivariance is preserved along the mean curvature flow. Here, we note that isotropy group acts naturally on both and . If the flow is graphical, then it follows from the -equivariance of the flow that it can be described by using -invariant functions on . We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that is a rank one symmetric space of compact type and the warping function satisfies certain additional properties. The proof is carried out by estimating the gradient of the -invariant functions satisfying the flow equation.
Comments: 16 pages, 2 figures
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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6C.8Monotonicity, ε-Regularity and Tangent Cones
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.