Regularity of minimal submanifolds and mean curvature flows with a common free boundary

Brian Krummel

Paper · Sep 26, 2016

Regularity of minimal submanifolds and mean curvature flows with a common free boundary

Brian Krummel

math.DGWider flowsarXiv:1609.08036

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Abstract

Let $N$ be a smooth $(n + l)$ -dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1, μ}$ $n$ -dimensional submanifolds-with-boundary $M_{k} \subset N$ with a common boundary $Γ$ , then $Γ$ is smooth and each $M_{k}$ is smooth up to $Γ$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l = 1$ . We additionally show that if ${V_{t}}_{t \in (- 1, 1)}$ is a Brakke flow such that each time-slice $V_{t}$ is a union of three or more $n$ -dimensional submanifolds-with-boundary $M_{k, t} \subset N$ with a common boundary $Γ_{t}$ and with parabolic $C^{2 + μ}$ regularity in time-space, then ${Γ_{t}}_{t \in (- 1, 1)}$ and ${M_{k, t}}_{t \in (- 1, 1)}$ are smooth (second Gevrey with real-analytic time-slices in the case $N$ is real-analytic).

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Submitted: Sep 26, 2016
Subjects: math.DG, math.AP

Comments: 36 pages

Mean curvature flow Weak & singular flows Boundaries & non-compact Free boundary & capillary Minimal & CMC surfaces

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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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.
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] 12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.

[2] 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.

Regularity of minimal submanifolds and mean curvature flows with a common free boundary

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