Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

Haizhong Li; Ruixuan Li; Changwei Xiong

Paper · Jun 15, 2023

Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

Haizhong Li, Ruixuan Li, Changwei Xiong

math.DGWider flowsarXiv:2306.09009

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Abstract

We derive various sharp upper bounds for the $p$ -capacity of a smooth compact set $K$ in the hyperbolic space $H^{n}$ and the Euclidean space $R^{n}$ . Firstly, using the inverse mean curvature flow, for the mean convex and star-shaped set $K$ in $H^{n}$ , we obtain sharp upper bounds for the $p$ -capacity $Cap_{p} (K)$ in three cases: (1) $n \geq 2$ and $p = 2$ , (2) $n = 2$ and $p \geq 3$ , (3) $n = 3$ and $1 < p \leq 3$ ; Using the unit-speed normal flow, we prove a sharp upper bound for $Cap_{p} (K)$ of a convex set $K$ in $H^{n}$ for $n \geq 2$ and $p > 1$ . Secondly, for the compact set $K$ in $R^{3}$ , using the weak inverse mean curvature flow, we get a sharp upper bound for the $p$ -capacity ( $1 < p < 3$ ) of the set $K$ with connected boundary; Using the inverse anisotropic mean curvature flow, we deduce a sharp upper bound for the anisotropic $p$ -capacity ( $1 < p < 3$ ) of an $F$ -mean convex and star-shaped set $K$ in $R^{3}$ .

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Submitted: Jun 15, 2023
Subjects: math.DG

Comments: 23 pages; all comments are welcome. arXiv admin note: text overlap with arXiv:2104.09905

Mean curvature flow Other curvature flows Inverse mean curvature flow Anisotropic & nonlocal flows

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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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978)

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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.

[1] 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.

Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

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