Gap Results for Free Boundary CMC Surfaces in Radially Symmetric Conformally Euclidean Three-Balls
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Gap Results for Free Boundary CMC Surfaces in Radially Symmetric Conformally Euclidean Three-Balls Maria Andrade1,2
· Ezequiel Barbosa2 · Edno Pereira2
Received: 5 June 2020 / Accepted: 11 November 2020 © Mathematica Josephina, Inc. 2020
Abstract ¯ as the Euclidean three-ball with radius r In this work, we consider M = (Br3 , g) equipped with the metric g¯ = e2h , conformal to the Euclidean metric, where the function h = h(x) depends only on the distance of x to the center of Br3 . We show that if a free boundary CMC surface Σ in M satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of Σ, the positional conformal vector field x and its potential function σ, then either Σ is a disk or Σ is an annulus rotationally symmetric. These results extend to the CMC case and to many other different conformally Euclidean spaces the main result obtained by Li and Xiong (J Geom Anal 28(4):3171–3182, 2018). Keywords Gap theorem · Constant mean curvature surfaces · Free boundary · Conformally Euclidean spaces Mathematics Subject Classification 53C20 · 53A10 · 49Q10
The authors were partially supported by PNPD/CAPES, CNPq, and FAPEMIG/Brazil agencies Grants.
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Maria Andrade [email protected] Ezequiel Barbosa [email protected] Edno Pereira [email protected]
1
Departamento de Matemática, Universidade Federal de Sergipe (UFS), São Cristóvão, SE 49100-000, Brazil
2
Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, Belo Horizonte, MG 30123-970, Brazil
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M. Andrade et al.
1 Introduction Let M be a three-dimensional Riemannian manifold with smooth boundary ∂ M. Let x : Σ → M be an isometric immersion, where Σ is a smooth compact surface with ∂Σ ⊆ ∂ M. As is well known, Σ is a free boundary CMC surface, if the mean curvature is constant and Σ is orthogonal to ∂ M at every point of ∂Σ. The first variation formula shows that free boundary CMC surfaces are critical points of the area functional for volume preserving variations of Σ, whose ∂Σ is free to move in ∂ M. In the situation where a minimal surface lies in a three-dimensional Euclidean unit ball B3 with free boundary, the flat equatorial disk and the critical catenoid are the best known examples. Here the critical catenoid is a piece of a catenoid in R3 which intersects ∂B3 orthogonally. In the setting where a CMC surface, with nonzero mean curvature, lies in a three-dimensional Euclidean unit ball B3 with free boundary, the spherical caps and the pieces of Delaunay surfaces, inside the ball which intersects ∂B3 orthogonally, are the best known examples. Many abstract examples of free boundary minimal surfaces in the unit three-dimensional ball were recently constructed by using the desingularization method or the gluing method (see, for instance, [5,7–9]), and it is expected these methods can also be used to build other examples of free boundary CMC surfaces with high genus and multiple boundary components. In this work, we are interested in the classifi
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