Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Shu-Yu Hsu

Paper · Jan 18, 2020

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Shu-Yu Hsu

math.APWider flowsarXiv:2001.06596

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Abstract

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf.) 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 $\frac{1}{n} < λ < \frac{1}{n - 1}$ and $μ < 0$ , we will give a new proof of the existence of a unique solution $r (y) \in C^{2} (μ, \infty) \cap C ([μ, \infty))$ of the equation $\frac{r _{y y} ( y )}{1 + r _{y} ( y ) ^{2}} = \frac{n - 1}{r ( y )} - \frac{1 + r _{y} ( y ) ^{2}}{λ ( r ( y ) - y r _{y} ( y ))}$ , $r (y) > 0$ , in $(μ, \infty)$ which satisfies $r (μ) = 0$ and $r_{y} (μ) = lim_{y ↘ μ} r_{y} (y) = + \infty$ . We also prove that there exist constants $y_{2} > y_{1} > 0$ such that $r_{y} (y) > 0$ for any $μ < y < y_{1}$ , $r_{y} (y_{1}) = 0$ , $r_{y} (y) < 0$ for any $y > y_{1}$ , $r_{y y} (y) < 0$ for any $μ < y < y_{2}$ , $r_{y y} (y_{2}) = 0$ and $r_{y y} (y) > 0$ for any $y > y_{2}$ . Moreover $lim_{y \to + \infty} r (y) = 0$ and $lim_{y \to + \infty} y r_{y} (y) = 0$ .

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Submitted: Jan 18, 2020
Subjects: math.AP, math.DG

Comments: 14 pages

Mean curvature flow Other curvature flows Inverse mean curvature flow 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 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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[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.

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

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