Constant mean curvature surfaces in sub-Riemannian geometry

By Robert K. Hladky, Scott D. Pauls. 2005

83 citations recorded by the source corpus.

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, $II_0$, for vertically rigid spaces and, as a first application, use $II_0$ to show that minimal surfaces cannot have points of horizontal positive curvature and, that minimal surfaces in Carnot groups cannot be locally horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

Subfields: Applied Mathematics, Genetics

View Paper: https://www.semanticscholar.org/paper/5efa67cbd432325c30097b6906a082b8fb94968c

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In Genetics

Citations in the corpus

Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “To quote a few: H0 can be defined in terms of the first variation of the area functional [20, 29, 9, 36, 40, 31, 7], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures H" in the Riemannian metrics g" as " → 0. It is also well known (see f or example [9, 29, 18”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, background): “Several first variation formula have been independently established in receent years, see for instance [18], [7], [8], [4], [32], [33], [27], [28], [35], [36].”
The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, methodology): “In this section we want to obtain first and second variation formulas of the area functional for intrinsic graphs; similar formulas have been obtained independently in [16,34] for more general surfaces.”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, methodology): “When specialized to Carnot groups, the adapted connection in [56] coincides with the horizontal connection in Definition 5.”