Abstract
We establish the second variation of sub-Riemannian surface measure for minimal non-horizontal submanifolds of a sub-Riemannian stratified Lie group. We obtain some applications for codimension one. Furthermore, we present a new proof of the fact that the hyperbolic paraboloid is stable in the Heisenberg group.
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References
Agrachev A, Barilari D, Boscain U. On the Hausdorff volume in sub-Riemannian geometry. Calc Var 2012;43:355–88.
Danielli D, Garofalo N, Nhieu DM. Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv Math 2007;215:292–378.
Diniz MM, Santos MRB, Veloso JMM. First variation of the Hausdorff measure of non-horizontal submanifolds in sub-Riemanniann stratified lie Groups. J Dynam Control Syst 2017;23:509–33.
Galli M. First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-Hermitian manifolds. Calc Var 2011;47:117–57.
Hladky R, Pauls SD. Variation of perimeter measure in sub-Riemannian geometry. Int Electron J Geom 2013;6:8–40.
Hurtado A, Ritoré M, Rosales C. The classification of complete stable area-stationary surfaces in the Heisenberg group H1. Adv Math 2010;224:561–600.
Montefalcone F. Hypersurfaces and variational formulas in sub-Riemannian Carnot groups. Journal de mathématiques pures et appliquées 2007;87:453–94.
Montefalcone F. Stable H-minimal hypersurfaces. J Geom Anal 2015; 25(2):820–70.
Pauls SD. Minimal surfaces in the Heisenberg group. Geom Dedicata 2004;104(1):201–31.
Ritoré M, Rosales C. Area-stationary surfaces in the Heisenberg group H1. Adv Math 2008;219(2):633–71.
Bernstein S. Sur un thééorème de géométrie et ses applications aux équations aux déricées partielles du type elliptique. Comm. de la Soc Math de Kharkov 1915;15:38–45.
Danielli D, Garofalo N, Nhieu DM, Pauls SD. Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1. J Differential Geom 2009;81(2):251–95.
Danielli D, Garofalo N, Nhieu DM, Pauls SD. The Bernstein Problem for Embedded Surfaces in the Heisenberg Group H1. Indiana University Mathematics Journal 2010;563–594.
Nicolussi S, Serra Cassano F. The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg group. Calc Var 2019;58(4):1–28.
Monti R, Serra Cassano F, Vittone D. A negative answer to the Bernstein problem for the Intrinsic Graphs in the Heisenberg Group. Bollettino Della Unione Matematica Italiana 2008;1:709–28.
Magnani V, Vittone D. An intrinsic measure for submanifolds in stratified groups. J Reine Angew Math 2018;619:203–32.
Xin YL. Minimal submanifolds and related topics. World Scientific; 1979.
Cheng JH, Hwang JF, Malchiodi A, Yang P. Minimal surfaces in pseudohermitian geometry. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 2005;4(1):129–77.
Spivak M. A comprehensive introduction to differential geometry, vol. II, 3rd ed. Houston: Publish or Perish, Inc.; 1999.
Spivak M. A comprehensive introduction to differential geometry, vol. IV, 2nd ed. Publish or: Perish Inc.; 1979.
Do Carmo MP, Flaherty Francis J. 1992. Riemannian geometry, vol 6. Springer.
Kobayashi S, Nomizu K. 1963. Foundations of differential geometry, vol. 1. Interscience.
Funding
This work has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant 428299/2018-0.
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José M. M. Veloso contributed equally to this work.
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Santos, M.R.B., Veloso, J.M.M. Second Variation of Sub-Riemannian Surface Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups. J Dyn Control Syst 29, 721–756 (2023). https://doi.org/10.1007/s10883-022-09606-0
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DOI: https://doi.org/10.1007/s10883-022-09606-0
Keywords
- Second variation
- Sub-Riemannian measure
- Non-horizontal submanifolds
- Stratified groups
- Heisenberg group
- Minimal surfaces
- Stability