Existence of hypercylinder expanders of the inverse mean curvature flow

K. M. Hui

Paper · Mar 19, 2018

Existence of hypercylinder expanders of the inverse mean curvature flow

K. M. Hui

math.AParXiv:1803.07425v3

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Abstract

We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $R^{n} \times R$ , $n \geq 2$ , of the form $(r, y (r))$ or $(r (y), y)$ where $r = ∣ x ∣$ , $x \in R^{n}$ , is the radially symmetric coordinate and $y \in R$ . More precisely for any $λ > \frac{1}{n - 1}$ and $μ > 0$ , we will give a new proof of the existence of a unique even solution $r (y)$ of the equation $\frac{r ^{''} ( y )}{1 + r ^{'} ( y ) ^{2}} = \frac{n - 1}{r ( y )} - \frac{1 + r ^{'} ( y ) ^{2}}{λ ( r ( y ) - y r ^{'} ( y ))}$ in $R$ which satisfies $r (0) = μ$ , $r^{'} (0) = 0$ and $r (y) > y r^{'} (y) > 0$ for any $y \in R$ . We will prove that $lim_{y \to \infty} r (y) = \infty$ and $a_{1} := lim_{y \to \infty} r^{'} (y)$ exists with $0 \leq a_{1} < \infty$ . We will also give a new proof of the existence of a constant $y_{1} > 0$ such that $r^{''} (y_{1}) = 0$ , $r^{''} (y) > 0$ for any $0 < y < y_{1}$ and $r^{''} (y) < 0$ for any $y > y_{1}$ .

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Submitted: Mar 19, 2018
Last revised: Dec 21, 2019 (v3)
Subjects: math.AP, math.DG

Comments: 12 pages, some proofs changed, reference updated

Mean curvature flow Other curvature flows Inverse mean curvature flow Ricci solitons 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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
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] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[3] 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 hypercylinder expanders of the inverse mean curvature flow

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