Optimal constant problem in the $ {L}^{2}$ extension theorem

Guan, Qiʼan; Zhou, Xiangyu

doi:10.1016/j.crma.2012.08.007

Complex Analysis

Optimal constant problem in the

L^{2}

extension theorem
[Problème de la constante optimale dans le théorème dʼextension

L^{2}

]

Présenté par : Jean-Pierre Demailly

Guan, Qiʼan ¹ ; Zhou, Xiangyu ²

¹ Beijing International Center for Mathematical Research, Peking University, Beijing, China
² Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China

Comptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 753-756.

Résumé
Abstract

Dans cette Note, nous résolvons le problème de la détermination de la constante optimale dans le théorème dʼextension $L^{2}$ avec poids négligeable sur les variétés de Stein. En application, nous prouvons la conjecture de Suita sur des surfaces de Riemann arbitraires.

In this Note, we solve the optimal constant problem in the $L^{2}$ -extension theorem with negligible weight on Stein manifolds. As an application, we prove the Suita conjecture on arbitrary open Riemann surfaces.

Reçu le : 2012-07-04
Accepté le : 2012-08-29
Publié le : 2012-08-01

DOI : 10.1016/j.crma.2012.08.007

Affiliations des auteurs :

Guan, Qiʼan ¹ ; Zhou, Xiangyu ²

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Guan, Qiʼan; Zhou, Xiangyu. Optimal constant problem in the $ {L}^{2}$ extension theorem. Comptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 753-756. doi : 10.1016/j.crma.2012.08.007. http://www.numdam.org/articles/10.1016/j.crma.2012.08.007/

Bibliographie
Cité par

[1] Berndtsson, B. The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman, Ann. LʼInst. Fourier (Grenoble), Volume 46 (1996) no. 4, pp. 1083-1094

[2] Z. Blocki, On the Ohsawa–Takegoshi extension theorem, preprint, 2012.

[3] Z. Blocki, Suita conjecture and the Ohsawa–Takegoshi extension theorem, preprint, 2012.

[4] Cao, J.; Shaw, M.-C.; Wang, L. Estimates for the $\bar{\partial}$ -Neumann problem and nonexistence of $C^{2}$ Levi-flat hypersurfaces in ${CP}^{n}$ , Math. Z., Volume 248 (2004), pp. 183-221

[5] Chen, B. A remark on an extension theorem of Ohsawa, Chin. Ann. Math., Ser. A, Volume 24 (2003), pp. 129-134 (in Chinese)

[6] Chen, S.-C.; Shaw, M.-C. Partial Differential Equations in Several Complex Variables, AMS/IP, 2001

[7] Demailly, J.-P. On the Ohsawa–Takegoshi–Manivel $L^{2}$ extension theorem, September 1997, Paris (Progress in Mathematics) (2000)

[8] Demailly, J.-P. Complex analytic and differential geometry http://www-fourier.ujf-grenoble.fr/~demailly/books.html

[9] Demailly, J.-P. Analytic Methods in Algebraic Geometry, Higher Education Press, Beijing, 2010

[10] Guan, Q.A.; Zhou, X.Y.; Zhu, L.F. On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the twisted Bochner–Kodaira identity, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 797-800

[11] Manivel, L. Un théorème de prolongement $L^{2}$ de sections holomorphes dʼun fibré vectoriel, Math. Zeitschrift, Volume 212 (1993), pp. 107-122

[12] McNeal, J. On large values of $L^{2}$ holomorphic functions, Math. Res. Lett., Volume 3 (1996) no. 2, pp. 247-259

[13] McNeal, J.; Varolin, D. Analytic inversion of adjunction: $L^{2}$ extension theorems with gain, Ann. LʼInst. Fourier (Grenoble), Volume 57 (2007) no. 3, pp. 703-718

[14] Ohsawa, T. On the extension of $L^{2}$ holomorphic functions. III. Negligible weights, Math. Z., Volume 219 (1995) no. 2, pp. 215-225

[15] Ohsawa, T. Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J., Volume 137 (1995), pp. 145-148

[16] Ohsawa, T.; Takegoshi, K. On the extension of $L^{2}$ holomorphic functions, Math. Z., Volume 195 (1987), pp. 197-204

[17] Siu, Y.-T. Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geometry, Volume 17 (1982), pp. 55-138

[18] Siu, Y.-T. The Fujita conjecture and the extension theorem of Ohsawa–Takegoshi, Hayama, World Scientific (1996), pp. 577-592

[19] Siu, Y.-T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Göttingen, 2000, Springer, Berlin (2002), pp. 223-277

[20] Suita, N. Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., Volume 46 (1972), pp. 212-217

[21] Straube, E. Lectures on the $L^{2}$ -Sobolev Theory of the $\bar{\partial}$ -Neumann Problem, ESI Lectures in Mathematics and Physics, European Mathematical Society, Zürich, 2010

[22] Zhu, L.F.; Guan, Q.A.; Zhou, X.Y. On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the twisted Bochner–Kodaira identity with a non-smooth twist factor, J. Math. Pures Appl., Volume 97 (2012) no. 6, pp. 579-601

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