On transformations of analytic CR-structures

A. B. Sukhov

Izvestiya. Mathematics , Tome 67 (2003) no. 2, pp. 303-332.

Résumé

We establish a link between the CR-geometry of real-analytic submanifolds in $\mathbb C^n$ and the geometry of differential equations. The idea of our approach is to regard biholomorphisms of a Levi-non-degenerate real-analytic CR-manifold $\mathscr M$ as point Lie symmetries of the second-order holomorphic system of differential equations defining the Segre family of $\mathscr M$. This enables us to study the biholomorphism group of $\mathscr M$ by means of the geometric theory of differential equations. We give several examples and applications to CR-geometry: results on the finite-dimensionality of the biholomorphism group and precise estimates of its dimension, and an explicit parametrization of the Lie algebra of infinitesimal automorphisms.

Bibliographie

[1] Abrosimov A. V., “Opisanie lokalno bigolomorfnykh avtomorfizmov standartnykh kvadrik korazmernosti dva”, Matem. sb., 184:10 (1993), 3–52 | Zbl

[2] Baouendi M. S., Ebenfelt P., Rothschild L. P., “$\mathrm{CR}$-automorphisms of real analytic $\mathrm{CR}$ manifolds in complex space”, Comm. Anal. Geom., 6 (1998), 291–315 | MR | Zbl

[3] Bedford E., Pinchuk S. I., “Vypuklye oblasti s nekompaktnymi gruppami avtomorfizmov”, Matem. sb., 185:5 (1994), 3–26 | MR | Zbl

[4] Beloshapka V. K., “Teorema edinstvennosti dlya avtomorfizmov nevyrozhdennoi poverkhnosti v kompleksnom prostranstve”, Matem. zametki, 47:3 (1990), 17–22 | MR | Zbl

[5] Beloshapka V. K., “O golomorfnykh preobrazovaniyakh kvadriki”, Matem. sb., 182:2 (1991), 203–219 | MR | Zbl

[6] Bluman G. W., Kumei S., Symmetries and differential equations, Springer-Verlag, N.Y., 1989 | MR | Zbl

[7] Bokhner S., Martin U. T., Funktsii mnogikh kompleksnykh peremennykh, IL, M., 1951

[8] Cartan E., “Sur la géométrie pseudoconforme des hypersurfaces de deux variables complexes”, Ann. Math. Pura Apll., 11 (1932), 17–90 | DOI | MR | Zbl

[9] Chern S. S., Moser J. K., “Real hypersurfaces in complex manifolds”, Acta Math., 133 (1974), 219–271 | DOI | MR

[10] Chern S. S., “On the projective structure of a real hypersurface in $\mathbb C^{n+1}$”, Math. Scand., 36 (1975), 74–82 | MR | Zbl

[11] Coupet B., Meylan F., Sukhov A., “Holomorphic maps of algebraic $\mathrm{CR}$ manifolds”, Internat. Math. Res. Notices, 1999, no. 1, 1–29 | DOI | MR | Zbl

[12] Dickson L. E., “Differential equations from the group standpoint”, Ann. Math., 25 (1924), 287–378 | DOI | MR | Zbl

[13] Faran J., “Segre families and real hypersurfaces”, Invent. Math., 60 (1980), 135–172 | DOI | MR | Zbl

[14] Kuranishi M., “On E.Cartan's prolongation theorem of exterior differential systems”, Amer. J. Math., 86 (1964), 379–391 | DOI | MR | Zbl

[15] Lewy H., “On the boundary behavior of holomorphic mappings”, Rend. Acad. Lincei, 35 (1977), 1–8

[16] Lie S., Scheffers G., Vorlesungen uber Continuierliche Gruppen, Chelsea, N. Y., 1971 | MR | Zbl

[17] Loboda A. V., “Veschestvenno-analiticheskie porozhdayuschie podmnogoobraziya korazmernosti 2 v $\mathbb C^4$ i ikh bigolomorfnye otobrazheniya”, Izv. AN SSSR. Ser. matem., 52:5 (1988), 970–990 | Zbl

[18] Malgrange B., “Equation de Lie. I; II”, J. Diff.Geom., 6 (1972), 503–522 ; 7, 117–141 | MR | Zbl | MR

[19] Malgrange B., “L'involutivité générique des systèmes différentiels analytiques”, C. R. Acad. Sci. Paris. Sér. I. Math., 326 (1998), 863–866 | MR | Zbl

[20] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 | MR | Zbl

[21] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[22] Pinchuk S. I., “Ob analiticheskom prodolzhenii golomorfnykh otobrazhenii”, Matem. sb., 98:3 (1975), 416–436 | MR

[23] Pommare Zh.-F., Sistemy uravnenii s chastnymi proizvodnymi i psevdogruppy Li, Mir, M., 1983 | MR | Zbl

[24] Segre B., “Intorno al problema di Poincaré della rappresentazione pseudo-conforme”, Rend. Acc. Lincei, 13 (1931), 676–683 | Zbl

[25] Shiffman B., “Projective geometry and Poincare's theorem on automorphisms of the ball”, Enseign. Math., 41 (1995), 201–216 | MR

[26] Sharipov R., Sukhov A., “On $\mathrm{CR}$ mappings between algebraic Cauchy–Riemann manifolds and separate algebraicity for holomorphic functions”, Trans. AMS, 348 (1996), 767–780 | DOI | MR | Zbl

[27] Spencer D., “Overdetermined systems of linear partial differential equations”, Bull. AMS, 75 (1969), 179–239 | DOI | MR | Zbl

[28] Stanton N., “Infinitesimal $\mathrm{CR}$ automorphisms of real hypersurfaces”, Amer. J. Math., 118 (1996), 209–233 | DOI | MR | Zbl

[29] Sukhov A., “On $\mathrm{CR}$ mappings of real quadric manifolds”, Mich. Math. J., 41 (1994), 143–150 | DOI | MR | Zbl

[30] Sukhov A., “Segre varieties and Lie symmetries”, Math. Zeit., 238 (2001), 483–492 | DOI | MR | Zbl

[31] Sukhov A., “On maps of $\mathrm{CR}$ manifolds and transformations of differential equations”, C. R. Acad. Sci. Paris. Sér. I, 333 (2001), 545–550 | MR | Zbl

[32] Sukhov A., “$\mathrm{CR}$ maps and point Lie transformations”, Mich. Math. J., 50 (2002), 369–379 | DOI | MR | Zbl

[33] Sukhov A., Segre varieties, $\mathrm{CR}$ geometry and Lie symmetries of second order $PDE$ systems, Preprint, February 2000 http://xxx.lanl.gov/ps/math.CV/0002163

[34] Tresse A., Détermination des invariants ponctuels de l'équation differentielle du second ordre $y''=\omega(x,y,y')$, Hirzel, Leiptzig, 1896

[35] Tanaka N., “On infinitesimal automorphisms of Siegel domains”, J. Math. Soc. Japan, 22 (1970), 180–212 | MR | Zbl

[36] Tumanov A. E., “Konechnomernost gruppy $\mathrm{CR}$-avtomorfizmov standartnogo $\mathrm{CR}$-mnogoobraziya i sobstvennye golomorfnye otobrazheniya oblastei Zigelya”, Izv. AN SSSR. Ser. matem., 52:3 (1988), 651–659 | MR | Zbl

[37] Webster S. M., “On the mapping problem for algebraic real hypersurfaces”, Inv. Math., 43 (1977), 53–68 | DOI | MR | Zbl