Paper · Nov 12, 2024

Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

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Abstract

In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $- 1$ (including the inverse mean curvature flow) in warped product spaces $I \times_{φ} M^{n}$ , where $M^{n}$ is a compact homogeneous manifold and $φ^{''} \geq 0$ , must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $- 1$ and with an extra assumption $φ^{'} \geq 0$ . Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.

Comments: 20 pages

Mean curvature flow Other curvature flows Inverse mean curvature flow Self-shrinkers & translators Homogeneous spaces Stability & uniqueness Boundaries & non-compact Rotational symmetry & warped products

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

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