Abstract
We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields in ℝn. It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries. Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical hypersurfaces.
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References
M. Berger, Geometry I. Universitext, Springer-Verlag, Berlin 1987.
W. Beckner, On the Grushin operator and hyperbolic symmetry, Proceedings of the American Mathematical Society 129 (2001), 1233–1246.
T. Bieske and J. Gong, The P-Laplace equation on a class of Grushin-type spaces, Proceedings of the American Mathematical Society 134 (2006), 3585–3594.
L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Communications on Pure and Applied Mathematics 50 (1997), 867–889.
B. Franchi and E. Lanconelli, Une métrique associée á une classe d’opérateurs elliptiques dégénérés (French) [A metric associated with a class of degenerate elliptic operators], Conference on Linear Partial and Pseudodifferential Operators (Torino, 1982), Rendiconti del Seminario Matematico. Universitá e Politecnico Torino 1983, Special Issue, 105–114 (1984).
N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabetype equations on groups of Heisenberg type, Duke Mathematical Journal 106 (2001), 411–448.
J. Heinonen and P. Koskela, Definitions of quasiconformality. Inventiones Mathematicae 120 (1995), 61–79.
A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Inventiones Mathematicae 80 (1985), 309–338.
A. Korányi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Advances in Mathematics 111 (1995), 1–87.
W. Kühnel and H.-B Rademacher, Conformal diffeomorphisms preserving the Ricci tensor, Proceedings of the American Mathematical Society 123 (1995), 2841–2848.
R. S. Kulkarni, Conformal structures and Möbius structures, in Conformal Geometry, Bonn, 1985/1986, Aspects of Mathematics, E12, Vieweg, Braunschweig, 1988, pp. 1–39.
J. Lelong-Ferrand, Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms. Differential geometry and relativity, Mathematical Physics and Applied Mathematics, Vol. 3, Reidel, Dordrecht, 1976, pp. 91–105.
J. Liouville, Extension au case des trois dimensions de la question du tracé géographique, Note VI, in G. Monge, Application de l’Analyse á la géometrie, Bachelier, Paris 1850, pp. 609–615.
M. Listing, Conformal Einstein spaces in N-dimensions. II, Journal of Geometry and Physics 56 (2006), 386–404.
D. Lupo and K. Payne, Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types, Duke Mathematical Journal 127 (2005), 251–290.
D. Lupo and K. Payne, Critical exponents for semilinear equations of mixed elliptichyperbolic and degenerate types, Communications on Pure and Applied Mathematics 56 (2003), 403–424.
R. Monti, Sobolev inequalities for weighted gradients, Communications in Partial Differential Equations 31 (2006), 1479–1504.
R. Monti and D. Morbidelli, Kelvin transform for Grushin operators and critical semilinear equations, Duke Mathematical Journal 131 (2006), 167–202.
A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Mathematica 155 (1985), 103–147.
B. O’Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, cademic Press [Harcourt Brace Jovanovich, Publishers], New York, 1983.
B. Osgood and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Mathematical Journal 67 (1992), 57–99.
P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Annals of Mathematics (2) 129 (1989), 1–60.
K. R. Payne, Singular metrics and associated conformal groups underlying differential operators of mixed and degenerate types, Annali di Matematica Pura ed Applicata 185 (2006), 613–625.
H. M. Reimann, Rigidity of H-type groups, Mathematische Zeitschrift 237 (2001), 697–725.
H. M. Reimann and F. Ricci, The complexified Heisenberg group, Proceedings on Analysis and Geometry (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, pp. 465–480.
R. Schoen and S. T. Yau, Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994.
P. Tang, Regularity and extremality of quasiconformal homeomorphisms on CR 3-manifolds, Annales Academiæ Scientiarium Fennicæ 21 (1996), 289–308.
J. Tyson, Sharp weighted Young’s inequalities and Moser-Trudinger inequalities on groups of Heisenberg type and Grushin spaces, Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 24 (2006), 357–384.
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Morbidelli, D. Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics. Isr. J. Math. 173, 379–402 (2009). https://doi.org/10.1007/s11856-009-0097-7
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DOI: https://doi.org/10.1007/s11856-009-0097-7