Lewy’s curves and chains on real hypersurfaces
HTML articles powered by AMS MathViewer
- by James J. Faran PDF
- Trans. Amer. Math. Soc. 265 (1981), 97-109 Request permission
Abstract:
Lewy’s curves on an analytic real hypersurface $M = \{ r(z,z) = 0\}$ in ${{\mathbf {C}}^2}$ are the intersections of $M$ with any of the Segre hypersurfaces ${Q_w} = \{ z:r(z,w) = 0\}$. If $M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy’s curves are chains, $M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in ${{\mathbf {C}}^n}$, $n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.References
- Shiing Shen Chern, On the projective structure of a real hypersurface in $C_{n+1}$, Math. Scand. 36 (1975), 74–82. MR 379910, DOI 10.7146/math.scand.a-11563
- S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146 J. J. Faran, Segre families and real hypersurfaces, Thesis, University of California at Berkeley, 1978. B. Segre, I. Intoruo al problema di Poincaré della rappresentazione pseudo-conforme, Rend Acc. Lincei 13 (1931), 676-683; II. Questioni geometriche legate colla teoria delle funzioni di due variabili complesse, Rend. Semin. Mat. Roma 7 (1931). S. M. Webster, Real hypersurfaces in complex space, University of California at Berkeley, 1975.
- 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
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 265 (1981), 97-109
- MSC: Primary 32C05; Secondary 32F25
- DOI: https://doi.org/10.1090/S0002-9947-1981-0607109-0
- MathSciNet review: 607109