A short proof of the Dimension Conjecture for real hypersurfaces in ℂ²

Isaev, Alexander; Kruglikov, Boris

doi:10.1090/proc/13070

A short proof of the Dimension Conjecture for real hypersurfaces in $\mathbb {C}^2$
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Proc. Amer. Math. Soc. 144 (2016), 4395-4399 Request permission

Abstract:

Recently, I. Kossovskiy and R. Shafikov settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in $\mathbb {C}^2$ via the dimension of the algebra of infinitesimal CR-automorphisms. In this note, we propose a short argument for obtaining their result.

References

M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103, DOI 10.1515/9781400883967
V. K. Beloshapka, On the dimension of the group of automorphisms of an analytic hypersurface, Math. USSR, Izv. 14 (1980), 223–245.
V. K. Beloshapka, Holomorphic transformations of a quadric, Mat. Sb. 182 (1991), no. 2, 203–219 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 1, 189–205. MR 1103272, DOI 10.1070/SM1992v072n01ABEH001267
V. K. Beloshapka, Real submanifolds of a complex space: their polynomial models, automorphisms, and classification problems, Uspekhi Mat. Nauk 57 (2002), no. 1(342), 3–44 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 1, 1–41. MR 1914541, DOI 10.1070/RM2002v057n01ABEH000474
V. K. Beloshapka, Can a stabilizer be eight-dimensional?, Russ. J. Math. Phys. 19 (2012), no. 2, 135–145. MR 2926319, DOI 10.1134/S106192081202001X
Elie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl. 11 (1933), no. 1, 17–90 (French). MR 1553196, DOI 10.1007/BF02417822
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
Michael Eastwood and Alexander Isaev, Examples of unbounded homogeneous domains in complex space, Sci. China Ser. A 48 (2005), no. suppl., 248–261. MR 2156506, DOI 10.1007/BF02884711
V. V. Ezhov and A. V. Isaev, On the dimension of the stability group for a Levi non-degenerate hypersurface, Illinois J. Math. 49 (2005), no. 4, 1155–1169. MR 2210357
Victor W. Guillemin and Shlomo Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968), 110–116. MR 217226, DOI 10.1090/S0002-9947-1968-0217226-9
Alexander Isaev and Dmitri Zaitsev, Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute parallelisms, J. Geom. Anal. 23 (2013), no. 3, 1571–1605. MR 3078365, DOI 10.1007/s12220-013-9419-4
Martin Kolář and Bernhard Lamel, Holomorphic equivalence and nonlinear symmetries of ruled hypersurfaces in $\Bbb {C}^2$, J. Geom. Anal. 25 (2015), no. 2, 1240–1281. MR 3319970, DOI 10.1007/s12220-013-9465-y
Ilya Kossovskiy and Rasul Shafikov, Analytic differential equations and spherical real hypersurfaces, J. Differential Geom. 102 (2016), no. 1, 67–126. MR 3447087
Boris Kruglikov, Point classification of second order ODEs: Tresse classification revisited and beyond, Differential equations: geometry, symmetries and integrability, Abel Symp., vol. 5, Springer, Berlin, 2009, pp. 199–221. With an appendix by Kruglikov and V. Lychagin. MR 2562566, DOI 10.1007/978-3-642-00873-3_{1}0
B. Kruglikov and D. The, The gap phenomenon in parabolic geometries, to appear in J. reine angew. Math., DOI: 10.1515/crelle-2014-0072, http://arxiv.org/abs/1303.1307.
G. D. Mostow, On maximal subgroups of real Lie groups, Ann. of Math. (2) 74 (1961), 503–517. MR 142687, DOI 10.2307/1970295
H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Math. Palermo 23 (1907), 185–220.
Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR 591460
Nancy K. Stanton, Real hypersurfaces with no infinitesimal CR automorphisms, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 469–473. MR 1771288, DOI 10.1090/conm/251/03888
Noboru Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131–190. MR 589931, DOI 10.4099/math1924.2.131
A. Tresse, Détermination des Invariants Ponctuels de L’équation Différentielle Ordinaire du Second Ordre $y'' = \omega (x,y,y’)$, Preisschr. Fürstlich Jablon. Ges., Hirzel, Leipzig, 1896.