Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups

Capogna, Luca; Pauls, Scott; Tyson, Jeremy

doi:10.1090/S0002-9947-10-04768-9

Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups
HTML articles powered by AMS MathViewer

by Luca Capogna, Scott D. Pauls and Jeremy T. Tyson PDF

Trans. Amer. Math. Soc. 362 (2010), 4045-4062 Request permission

Abstract:

We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo–Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.

References

Nicola Arcozzi and Fausto Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z. 256 (2007), no. 3, 661–684. MR 2299576, DOI 10.1007/s00209-006-0098-8
Zoltán M. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math. 564 (2003), 63–83. MR 2021034, DOI 10.1515/crll.2003.094
Zoltán M. Balogh and Matthieu Rickly, Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 847–868. MR 2040646
O. Călin and V. Mangione, Variational calculus on sub-Riemannian manifolds, Balkan J. Geom. Appl. 8 (2003), no. 1, 21–32. Volume dedicated to the memory of Grigorios Tsagas (1935–2003), president of the Balkan Society of Geometers (1997–2003). MR 2030318
Ovidiu Calin and Vittorio Mangione, Geodesics with constraints on Heisenberg manifolds, Results Math. 44 (2003), no. 1-2, 44–53. MR 2011905, DOI 10.1007/BF03322912
Luca Capogna and Giovanna Citti, Generalized mean curvature flow in Carnot groups, Comm. Partial Differential Equations 34 (2009), no. 7-9, 937–956. MR 2560306, DOI 10.1080/03605300903050257
Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007. MR 2312336, DOI 10.1007/978-3-7643-8133-2
Danielli, D., Garofalo, N., and Nhieu, D.-M. Minimal surfaces, surfaces of constant mean curvature and isoperimetry in Carnot groups. preprint, 2001.
Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263–341. MR 2014879, DOI 10.4310/CAG.2003.v11.n2.a5
D. Danielli, N. Garofalo, and D. M. Nhieu, Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. Math. 215 (2007), no. 1, 292–378. MR 2354992, DOI 10.1016/j.aim.2007.04.004
Maklouf Derridj, Sur un théorème de traces, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 2, 73–83 (French, with English summary). MR 343011, DOI 10.5802/aif.413
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
Nicola Garofalo and Federico Tournier, New properties of convex functions in the Heisenberg group, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2011–2055. MR 2197446, DOI 10.1090/S0002-9947-05-04016-X
Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823
Cristian E. Gutiérrez and Annamaria Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations 29 (2004), no. 9-10, 1305–1334. MR 2103838, DOI 10.1081/PDE-200037752
Cristian E. Gutiérrez and Annamaria Montanari, On the second order derivatives of convex functions on the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 2, 349–366. MR 2075987
Robert K. Hladky and Scott D. Pauls, Constant mean curvature surfaces in sub-Riemannian geometry, J. Differential Geom. 79 (2008), no. 1, 111–139. MR 2401420
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Petri Juutinen, Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on Carnot groups, Rev. Mat. Iberoam. 23 (2007), no. 1, 191–200. MR 2351130, DOI 10.4171/RMI/490
Adam Korányi, Geometric aspects of analysis on the Heisenberg group, Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 209–258. MR 748865
Adam Korányi and Hans Martin Reimann, Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2) 111 (1987), no. 1, 3–21 (English, with French summary). MR 886958
Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 1–22. MR 2027845, DOI 10.1007/s00526-003-0190-4
Valentino Magnani, Lipschitz continuity, Aleksandrov theorem and characterizations for $H$-convex functions, Math. Ann. 334 (2006), no. 1, 199–233. MR 2208954, DOI 10.1007/s00208-005-0717-4
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Pierre Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 127–130 (French, with English summary). MR 676380
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
Changyou Wang, Viscosity convex functions on Carnot groups, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1247–1253. MR 2117228, DOI 10.1090/S0002-9939-04-07836-0