The Parabolic U(1)-Higgs Equations and Codimension-two Mean Curvature Flows

Davide Parise; Alessandro Pigati; Daniel Stern

Paper · Jul 27, 2023

The Parabolic $U (1)$ -Higgs Equations and Codimension-two Mean Curvature Flows

Davide Parise, Alessandro Pigati, Daniel Stern

math.DGWider flowsarXiv:2307.15134

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Abstract

We develop the asymptotic analysis as $ε \to 0$ for the natural gradient flow of the self-dual $U (1)$ -Higgs energies $E_{ε} (u, \nabla) = \int_{M} (∣\nabla u ∣^{2} + ε^{2} ∣ F_{\nabla} ∣^{2} + \frac{( 1 - ∣ u ∣ ^{2} ) ^{2}}{4 ε ^{2}})$ on Hermitian line bundles over closed manifolds $(M^{n}, g)$ of dimension $n \geq 3$ , showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows – i.e., integral $(n - 2)$ -Brakke flows – generalizing results of the last two authors from the stationary case. Given any integral $(n - 2)$ -cycle $Γ_{0}$ in $M$ , these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at $Γ_{0}$ with additional structure, similar to those produced via Ilmanen's elliptic regularization.

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Submitted: Jul 27, 2023
Subjects: math.DG

Comments: Comments welcome

Mean curvature flow Weak & singular flows

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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. 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)

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12C.9
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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
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[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.

The Parabolic $U (1)$ -Higgs Equations and Codimension-two Mean Curvature Flows

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