Examples of Ricci-mean curvature flows

Hikaru Yamamoto

Paper · Aug 17, 2016

Examples of Ricci-mean curvature flows

Hikaru Yamamoto

math.DGarXiv:1608.04944

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Abstract

Let $π : P (O (0) \oplus O (k)) \to P^{n - 1}$ be a projective bundle over $P^{n - 1}$ with $1 \leq k \leq n - 1$ . In this paper, we show that lens space $L (k; 1) (r)$ with radius $r$ embedded in $P (O (0) \oplus O (k))$ is a self-similar solution, where $P (O (0) \oplus O (k))$ is endowed with the $U (n)$ -invariant gradient shrinking Ricci soliton structure. We also prove that there exists a pair of critical radii $r_{1} < r_{2}$ which satisfies the following. The lens space $L (k; 1) (r)$ is a self-shrinker if $r < r_{2}$ and self-expander if $r_{2} < r$ , and the Ricci-mean curvature flow emanating from $L (k; 1) (r)$ collapses to the zero section of $π$ if $r < r_{1}$ and to the $\infty$ -section of $π$ if $r_{1} < r$ . This gives explicit examples of Ricci-mean curvature flows.

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Submitted: Aug 17, 2016
Subjects: math.DG

Comments: 18 pages, 1 figure, 1 table

Mean curvature flow Ricci solitons Self-shrinkers & translators

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) 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) 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)

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.

Examples of Ricci-mean curvature flows

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