Index estimates for surfaces with constant mean curvature in 3-dimensional manifolds
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Calculus of Variations
Index estimates for surfaces with constant mean curvature in 3-dimensional manifolds Nicolau S. Aiex1 · Han Hong2 Received: 11 February 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove index estimates for closed and free boundary CMC surfaces in certain 3-dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are able to prove that the index of a CMC surface in an arbitrary 3-manifold is bounded below by a linear function of its genus. Mathematics Subject Classification 53A10 · 49Q10 · 49R05
1 Introduction A closed hypersurface of constant mean curvature (CMC) may be variationally characterized as a critical point of the area functional under variations that preserve enclosed volume. In a similar way, a free boundary constant mean curvature (free boundary CMC) hypersurface is an extremal solution of the same problem where, in addition, the boundary is restricted inside a closed hypersurface. If such a hypersurface minimizes area for small perturbations then it is stable for the corresponding problem. For example, solutions to the isoperimetric problem, that is, the hypersurface with or without boundary that has least area for a fixed enclosed volume, are in particular stable CMC hypersurfaces. In [3,4] Barbosa–do Carmo and Barbosa–do Carmo–Eschenburg classify stable closed CMC hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces. A similar result was obtained by Souam [22] for stable free boundary CMC hypersurfaces in a hemisphere and more recently for closed CMC surfaces in S 2 × R and H2 × R. Other classification results for stable free boundary CMC surfaces were obtained by Ros–Vergasta [19] and later
Communicated by A. Neves.
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Nicolau S. Aiex [email protected] Han Hong [email protected]
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University of British Columbia, ESB 4118, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada
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University of British Columbia, AA 138, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada 0123456789().: V,-vol
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N. S. Aiex, H. Hong
improved by Nunes [16]. It is also natural to study CMC hypersurfaces of higher index. That is, those that have some small pertubations that decrease area with fixed enclosed volume. In [24] Torralbo-Urbano make use of isometric embeddings of homogenous 3-manifolds into Euclidean space to study stable closed CMC surfaces and the isoperimetric problem in Berger spheres. The authors also use coordinates of hamornic vector fields to construct test functions for the second variation of area. In the case of minimal surfaces there has been multiple results establishing a connection between the topology of the surface and its index. For example, do Carmo–Peng [11], FischerColbrie–Schoen [12] and Pogorelov [17] have independently proved that stable two-sided minimal surfaces in R3 are planes. In [18] Ros proves that the index of a minimal surface in R3 is bounded below by a linear function of its genus, which was later improved by Chodosh–Maxi
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