TL;DR
This paper develops a min-max theory for free boundary geodesics on Riemannian manifolds, introducing a modified curve shortening process to find geodesic segments with free boundary conditions.
Contribution
It introduces a new min-max framework and a modified Birkhoff curve shortening process for free boundary problems in Riemannian geometry.
Findings
Established a min-max theory for free boundary geodesics.
Developed a modified Birkhoff curve shortening process.
Achieved a strong min-max approximation result similar to Colding-Minicozzi.
Abstract
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong "Colding-Minicozzi" type min-max approximation result.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · 3D Shape Modeling and Analysis · Advanced Numerical Analysis Techniques