The equivalence theory for infinite type hypersurfaces in ℂ²

Ebenfelt, Peter; Kossovskiy, Ilya; Lamel, Bernhard

doi:10.1090/tran/8627

The equivalence theory for infinite type hypersurfaces in $\mathbb {C}^{2}$
HTML articles powered by AMS MathViewer

by Peter Ebenfelt, Ilya Kossovskiy and Bernhard Lamel PDF

Trans. Amer. Math. Soc. 375 (2022), 4019-4056 Request permission

Abstract:

We develop a classification theory for real-analytic hypersurfaces in $\mathbb {C}^{2}$ in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in $\mathbb {C}^{2}$ in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR-DS technique in CR-geometry.

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
George W. Bluman and Sukeyuki Kumei, Symmetries and differential equations, Applied Mathematical Sciences, vol. 81, Springer-Verlag, New York, 1989. MR 1006433, DOI 10.1007/978-1-4757-4307-4
Boele L. J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 517–540 (English, with English and French summaries). MR 1182640
Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333–354 (French). MR 1556687
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
Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II, Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333–354 (French). MR 1556687
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
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
Peter Ebenfelt, Bernhard Lamel, and Dmitri Zaitsev, Normal form for infinite type hypersurfaces in $\Bbb C^2$ with nonvanishing Levi form derivative, Doc. Math. 22 (2017), 165–190. MR 3609198
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
Robert Juhlin and Bernhard Lamel, On maps between nonminimal hypersurfaces, Math. Z. 273 (2013), no. 1-2, 515–537. MR 3010173, DOI 10.1007/s00209-012-1017-9
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
I. Kossovskiy and B. Lamel, New extension phenomena for solutions of tangential Cauchy-Riemann equations, Comm. Partial Differential Equations 41 (2016), no. 6, 925–951. MR 3509519, DOI 10.1080/03605302.2016.1180536
I. Kossovskiy and B. Lamel, On the analyticity of CR-diffeomorphisms, Amer. J. Math. 140 (2018), no. 1, 139–188. MR 3749192, DOI 10.1353/ajm.2018.0002
I. Kossovskiy, B. Lamel, and L. Stolovitch, Equivalence of three-dimensional Cauchy-Riemann manifolds and multisummability theory, Adv. Math. 397 (2022), Paper No. 108117, 42. MR 4377933, DOI 10.1016/j.aim.2021.108117
Ilya Kossovskiy and Rasul Shafikov, Analytic differential equations and spherical real hypersurfaces, J. Differential Geom. 102 (2016), no. 1, 67–126. MR 3447087
Ilya Kossovskiy and Rasul Shafikov, Divergent CR-equivalences and meromorphic differential equations, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2785–2819. MR 3576537, DOI 10.4171/JEMS/653
I. Kossovskiy and D. Zaitsev, Convergent normal form and canonical connection for hypersurfaces of finite type in $\Bbb {C}^2$, Adv. Math. 281 (2015), 670–705. MR 3366850, DOI 10.1016/j.aim.2015.06.001
Bernhard Lamel and Nordine Mir, Finite jet determination of local CR automorphisms through resolution of degeneracies, Asian J. Math. 11 (2007), no. 2, 201–216. MR 2328892, DOI 10.4310/AJM.2007.v11.n2.a3
Francine Meylan, A reflection principle in complex space for a class of hypersurfaces and mappings, Pacific J. Math. 169 (1995), no. 1, 135–160. MR 1346250
H Poincaré. Les fonctions analytiques de deux variables et la représentation conforme. Palermo Rend. 23 (1907), 185–220.
Beniamino Segre. Questioni geometriche legate colla teoria delle funzioni di due variabili complesse. Rend. Sem. Mat. Roma II. Ser. 7 (1932), no. 2, 59–107.
A. B. Sukhov, On transformations of analytic CR-structures, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 2, 101–132 (Russian, with Russian summary); English transl., Izv. Math. 67 (2003), no. 2, 303–332. MR 1972994, DOI 10.1070/IM2003v067n02ABEH000428
Alexandre Sukhov, Segre varieties and Lie symmetries, Math. Z. 238 (2001), no. 3, 483–492. MR 1869694, DOI 10.1007/s002090100262
Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan 14 (1962), 397–429. MR 145555, DOI 10.2969/jmsj/01440397
S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), no. 1, 53–68. MR 463482, DOI 10.1007/BF01390203