Holomorphic Maps of Levi-Degenerate Tube Hypersurfaces in $\mathbb C^3$

N. G. Kruzhilin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 183-193.

Résumé

Locally biholomorphic maps between 2-nondegenerate smooth real tube hypersurfaces in $\mathbb C^3$ with Levi form of rank $1$ are described. It is shown that, except for hypersurfaces that are locally equivalent to the boundary of the future tube, such maps must be affine. The proof uses the local holomorphic version of the fundamental theorem of projective geometry which was earlier proved by the author.

Détail

Mots-clés : tube hypersurface
Keywords: holomorphic map, Levi form, complex line.

Bibliographie

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

[2] Fels G., Kaup W., “Classification of Levi degenerate homogeneous CR-manifolds in dimension 5”, Acta math., 201:1 (2008), 1–82 | DOI | MR | Zbl

[3] Isaev A., “On the CR-curvature of Levi degenerate tube hypersurfaces”, Methods Appl. Anal., 23:4 (2016), 317–328 | MR | Zbl

[4] Isaev A., Zaitsev D., “Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute parallelisms”, J. Geom. Anal., 23:3 (2013), 1571–1605 | DOI | MR | Zbl

[5] Kaup W., Zaitsev D., “On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces”, J. Eur. Math. Soc., 8:3 (2006), 465–490 | DOI | MR | Zbl

[6] Klingen H., “Diskontinuierliche Gruppen in symmetrischen Räumen. I”, Math. Ann., 129 (1955), 345–369 | DOI | MR | Zbl

[7] Kolář M., Kossovskiy I., A complete normal form for everywhere Levi-degenerate hypersurfaces in $\mathbb C^3$, E-print, 2019, arXiv: 1905.05629 [math.CV]

[8] N. K. Kruzhilin, “On a local holomorphic version of the fundamental theorem of projective geometry”, Russ. Math. Surv., 74:6 (2019), 1123–1125 | DOI | MR | Zbl

[9] Medori C., Spiro A., “The equivalence problem for five-dimensional Levi degenerate CR manifolds”, Int. Math. Res. Not., 2014:20 (2014), 5602–5647 | DOI | MR | Zbl

[10] Merker J., Pocchiola S., “Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M^5\subset \mathbb C^3$ of constant Levi rank 1”, J. Geom. Anal., 30:3 (2020), 2689–2730, 3233–3242 | DOI | MR | Zbl

[11] Mok N., Yeung S.K., “Geometric realizations of uniformization of conjugates of Hermitian locally symmetric manifolds”, Complex analysis and geometry, ed. by V. Ancona, A. Silva, Plenum Press, New York, 1993, 253–270 | DOI | MR | Zbl

[12] Pocchiola S., Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M^5\in \mathbb C^3$ of constant Levi rank 1, E-print, 2013, arXiv: 1312.6400 [math.CV]

[13] Extrinsic Geometry of Convex Surfaces, Transl. Math. Monogr., 35, Am. Math. Soc., Providence, RI, 1973 | Zbl

[14] Porter C., Zelenko I., Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation, E-print, 2019, arXiv: 1704.03999v5 [math.DG]

[15] Shiffman B., “Synthetic projective geometry and Poincaré's theorem on automorphisms of the ball”, Enseign. math. Sér. 2, 41:3–4 (1995), 201–215 | MR | Zbl

[16] Tanaka N., “On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables”, J. Math. Soc. Japan, 14:4 (1962), 397–429 | DOI | MR | Zbl