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May 2008 Constant mean curvature surfaces in sub-Riemannian geometry
R. K. Hladky, S. D. Pauls
J. Differential Geom. 79(1): 111-139 (May 2008). DOI: 10.4310/jdg/1207834659

Abstract

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II0, for vertically rigid spaces and, as a first application, use II0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in Carnot groups cannot be locally strictly horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

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R. K. Hladky. S. D. Pauls. "Constant mean curvature surfaces in sub-Riemannian geometry." J. Differential Geom. 79 (1) 111 - 139, May 2008. https://doi.org/10.4310/jdg/1207834659

Information

Published: May 2008
First available in Project Euclid: 10 April 2008

zbMATH: 1156.53038
MathSciNet: MR2401420
Digital Object Identifier: 10.4310/jdg/1207834659

Rights: Copyright © 2008 Lehigh University

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Vol.79 • No. 1 • May 2008
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