Higher order symmetries of real hypersurfaces in ℂ³

Kolar, Martin; Meylan, Francine

doi:10.1090/proc/13090

Higher order symmetries of real hypersurfaces in $\mathbb C^3$
HTML articles powered by AMS MathViewer

by Martin Kolar and Francine Meylan PDF

Proc. Amer. Math. Soc. 144 (2016), 4807-4818 Request permission

Abstract:

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By results of Kolar, Meylan, and Zaitsev in 2014, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in $\mathbb C^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author for an AIM workshop in 2010.

References

M. S. Baouendi, Peter Ebenfelt, and Xiaojun Huang, Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics, Adv. Math. 219 (2008), no. 5, 1427–1445. MR 2458142, DOI 10.1016/j.aim.2008.07.001
E. Bedford and S. I. Pinchuk, Convex domains with noncompact groups of automorphisms, Mat. Sb. 185 (1994), no. 5, 3–26 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 82 (1995), no. 1, 1–20. MR 1275970, DOI 10.1070/SM1995v082n01ABEH003550
V. K. Beloshapka, V. V. Ezhov, and G. Shmal′ts, Holomorphic classification of four-dimensional surfaces in $\Bbb C^3$, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 3, 3–18 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 3, 413–427. MR 2432751, DOI 10.1070/IM2008v072n03ABEH002406
V. K. Beloshapka and I. G. Kossovskiy, Classification of homogeneous CR-manifolds in dimension 4, J. Math. Anal. Appl. 374 (2011), no. 2, 655–672. MR 2729251, DOI 10.1016/j.jmaa.2010.07.008
Thomas Bloom and Ian Graham, On “type” conditions for generic real submanifolds of $\textbf {C}^{n}$, Invent. Math. 40 (1977), no. 3, 217–243. MR 589930, DOI 10.1007/BF01425740
Thomas Bloom and Ian Graham, A geometric characterization of points of type $m$ on real submanifolds of $\textbf {C}^{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182. MR 492369
Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in $\textbf {C}^{n}$, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 2, 125–322. MR 684898, DOI 10.1090/S0273-0979-1983-15087-5
Cartan, E., Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I , Ann. Math. Pura Appl. 11 (1932), 17–90.
David Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), no. 3, 529–586. MR 769163, DOI 10.2307/1971087
David Catlin, Subelliptic estimates for the $\overline \partial$-Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), no. 1, 131–191. MR 898054, DOI 10.2307/1971347
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015
Peter Ebenfelt, New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy, J. Differential Geom. 50 (1998), no. 2, 207–247. MR 1684982
Peter Ebenfelt, Xiaojun Huang, and Dmitri Zaitsev, The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics, Amer. J. Math. 127 (2005), no. 1, 169–191. MR 2115664, DOI 10.1353/ajm.2005.0004
Peter Ebenfelt, Bernhard Lamel, and Dmitri Zaitsev, Degenerate real hypersurfaces in $\Bbb C^2$ with few automorphisms, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3241–3267. MR 2485425, DOI 10.1090/S0002-9947-09-04626-1
P. Ebenfelt, B. Lamel, and D. Zaitsev, Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case, Geom. Funct. Anal. 13 (2003), no. 3, 546–573. MR 1995799, DOI 10.1007/s00039-003-0422-y
Gregor Fels and Wilhelm Kaup, Classification of Levi degenerate homogeneous CR-manifolds in dimension 5, Acta Math. 201 (2008), no. 1, 1–82. MR 2448066, DOI 10.1007/s11511-008-0029-0
Xiaojun Huang and Wanke Yin, A Bishop surface with a vanishing Bishop invariant, Invent. Math. 176 (2009), no. 3, 461–520. MR 2501295, DOI 10.1007/s00222-008-0167-1
Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008. MR 2363178, DOI 10.1090/gsm/086
A. V. Isaev and S. G. Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999), no. 1, 1–38. MR 1706680, DOI 10.1006/aima.1998.1821
Howard Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, vol. 32, American Mathematical Society, Providence, RI, 1990. MR 1067341, DOI 10.1090/surv/032
J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
Martin Kolar, Francine Meylan, and Dmitri Zaitsev, Chern-Moser operators and polynomial models in CR geometry, Adv. Math. 263 (2014), 321–356. MR 3239141, DOI 10.1016/j.aim.2014.06.017
Martin Kolář and Francine Meylan, Infinitesimal CR automorphisms of hypersurfaces of finite type in $\Bbb C^2$, Arch. Math. (Brno) 47 (2011), no. 5, 367–375. MR 2876940, DOI 10.5817/am2017-5-255
Martin Kolář, The Catlin multitype and biholomorphic equivalence of models, Int. Math. Res. Not. IMRN 18 (2010), 3530–3548. MR 2725504, DOI 10.1093/imrn/rnq013
Martin Kolář, Normal forms for hypersurfaces of finite type in ${\Bbb C}^2$, Math. Res. Lett. 12 (2005), no. 5-6, 897–910. MR 2189248, DOI 10.4310/MRL.2005.v12.n6.a10
Martin Kolář and Francine Meylan, Chern-Moser operators and weighted jet determination problems, Geometric analysis of several complex variables and related topics, Contemp. Math., vol. 550, Amer. Math. Soc., Providence, RI, 2011, pp. 75–88. MR 2868555, DOI 10.1090/conm/550/10867
M. Kolář and F. Meylan, Automorphism groups of Levi degenerate hypersurfaces in $\mathbb C^3$, preprint
N. G. Kruzhilin and A. V. Loboda, Linearization of local automorphisms of pseudoconvex surfaces, Dokl. Akad. Nauk SSSR 271 (1983), no. 2, 280–282 (Russian). MR 718188
Poincaré, H., Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo 23 (1907), 185–220.
Nancy K. Stanton, Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118 (1996), no. 1, 209–233. MR 1375306, DOI 10.1353/ajm.1996.0005
A. G. Vitushkin, Real-analytic hypersurfaces of complex manifolds, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 3–31, 237 (Russian). MR 786085
S. M. Webster, On the Moser normal form at a non-umbilic point, Math. Ann. 233 (1978), no. 2, 97–102. MR 486511, DOI 10.1007/BF01421918