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The First Width of Non-negatively Curved Surfaces with Convex Boundary

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Abstract

In this paper, free boundary geodesic networks whose length realizes the first min–max width of the length functional are investigated. This functional acts on the space of relative flat 1-dimensional cycles modulo 2 in a compact surface with boundary. The widths are special critical values of the volume functional in some class of submanifolds which naturally arise in the Min–max Theory of Almgren and Pitts. The main result of this work concerns the existence of a geodesic network with a rather simple structure which realizes the first width of a surface with non-negative sectional curvature and strictly convex boundary. More precisely, it is either a simple geodesic meeting the boundary orthogonally, or a geodesic loop with vertex at a boundary point determining two equal angles with that boundary curve.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, 57072970, Brazil

    Sidney Donato

  2. Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, CE, 60455760, Brazil

    Rafael Montezuma

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SD was supported by CAPES-Brazil 001. RM was supported by CNPq and Instituto Serrapilheira, grant “New perspectives of the min–max theory for the area functional”.

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Donato, S., Montezuma, R. The First Width of Non-negatively Curved Surfaces with Convex Boundary. J Geom Anal 34, 60 (2024). https://doi.org/10.1007/s12220-023-01511-7

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