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January 2021 Allen–Cahn min-max on surfaces
Christos Mantoulidis
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J. Differential Geom. 117(1): 93-135 (January 2021). DOI: 10.4310/jdg/1609902018

Abstract

We use a min-max procedure on the Allen–Cahn energy functional to construct geodesics on closed, $2$‑dimensional Riemannian manifolds, as motivated by the work of Guaraco [Gua18]. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen–Cahn curvature estimates due to Wang–Wei [WW19], we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding “entire” singularity models constructed by del Pino–Kowalczyk–Pacard [dPKP13] with Morse index $1$. The argument is completed by a Morse index estimate on these singularity models.

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Christos Mantoulidis. "Allen–Cahn min-max on surfaces." J. Differential Geom. 117 (1) 93 - 135, January 2021. https://doi.org/10.4310/jdg/1609902018

Information

Received: 20 August 2017; Published: January 2021
First available in Project Euclid: 6 January 2021

MathSciNet: MR4195753
Digital Object Identifier: 10.4310/jdg/1609902018

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 1 • January 2021
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