Existence of self-similar solution of the inverse mean curvature flow

K. M. Hui

Paper · Jan 25, 2018

Existence of self-similar solution of the inverse mean curvature flow

K. M. Hui

math.APWider flowsarXiv:1801.08250v4

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Abstract

We will give a new proof of a recent result of P. Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $R^{n}$ , $n \geq 2$ , of the form $u (x, t) = e^{λ t} f (e^{- λ t} x)$ for any constants $λ > \frac{1}{n - 1}$ and $μ < 0$ such that $f (0) = μ$ . More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $\mbox d i v (\frac{\nabla f}{1 + ∣\nabla f ∣ ^{2}}) = \frac{1}{λ} \cdot \frac{1 + ∣\nabla f ∣ ^{2}}{x \cdot \nabla f - f}$ in $R^{n}$ , $f (0) = μ$ , for any $λ > \frac{1}{n - 1}$ and $μ < 0$ , which satisfies $f_{r} (r) > 0$ , $f_{r r} (r) > 0$ and $r f_{r} (r) > f (r)$ for all $r > 0$ . We will also prove that $lim_{r \to \infty} \frac{r f _{r} ( r )}{f ( r )} = \frac{λ ( n - 1 )}{λ ( n - 1 ) - 1}$ .

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Submitted: Jan 25, 2018
Last revised: Aug 23, 2018 (v4)
Subjects: math.AP, math.DG

Comments: 21 pages, proof of Theorem 1.2 completely rewritten

Mean curvature flow Other curvature flows Inverse mean curvature flow 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 in the L² sense. 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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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.

Existence of self-similar solution of the inverse mean curvature flow

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