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Ohsawa–Takegoshi L2 extension theorem
From Handwiki - Reading time: 2 minShort description: Result concerning the holomorphic extensions In several complex variables
In several complex variables, the Ohsawa–Takegoshi L2 extension theorem is a fundamental result concerning the holomorphic extension of an [math]\displaystyle{ L^2 }[/math]-holomorphic function defined on a bounded Stein manifold (such as a pseudoconvex compact set in [math]\displaystyle{ \mathbb{C}^n }[/math] of dimension less than [math]\displaystyle{ n }[/math]) to a domain of higher dimension, with a bound on the growth. It was discovered by Takeo Ohsawa and Kensho Takegoshi in 1987,[1] using what have been described as ad hoc methods involving twisted Laplace–Beltrami operators, but simpler proofs have since been discovered.[2] Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.
See also
note
References
- Błocki, Zbigniew (2014). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences 4 (3): 433–480. doi:10.1007/s13373-014-0058-2.
- Demailly, Jean-Pierre (2000). "On the Ohsawa–Takegoshi–Manivel L2 extension theorem". Complex Analysis and Geometry. pp. 47–82. doi:10.1007/978-3-0348-8436-5_3. ISBN 978-3-0348-9566-8. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/ohsawa_tak.pdf.
- Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an [math]\displaystyle{ L^2 }[/math] extension problem with an optimal estimate and applications". Annals of Mathematics 181 (3): 1139–1208. doi:10.4007/annals.2015.181.3.6.
- Hörmander, Lars (1965). "L2 estimates and existence theorems for the [math]\displaystyle{ \bar \partial }[/math] operator". Acta Mathematica 113: 89–152. doi:10.1007/BF02391775.
- Ohsawa, Takeo; Takegoshi, Kensho (1987). "On the extension of [math]\displaystyle{ L^2 }[/math] holomorphic functions". Mathematische Zeitschrift 195 (2): 197–204. doi:10.1007/BF01166457.
- Ohsawa, Takeo (2017). "On the extension of [math]\displaystyle{ L^2 }[/math] holomorphic functions VIII — a remark on a theorem of Guan and Zhou". International Journal of Mathematics 28 (9). doi:10.1142/S0129167X17400055.
- Ohsawa, Takeo (10 December 2018). [math]\displaystyle{ L^2 }[/math] Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds. Springer Monographs in Mathematics. doi:10.1007/978-4-431-55747-0. ISBN 9784431568513.
- Bousfield Classes and Ohkawa's Theorem. Springer Proceedings in Mathematics & Statistics. 309. 2020. doi:10.1007/978-981-15-1588-0. ISBN 978-981-15-1587-3. https://books.google.com/books?id=ssPXDwAAQBAJ&pg=PA426.
- Siu, Yum-Tong (2011). "Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles". Science China Mathematics 54 (8): 1767–1802. doi:10.1007/s11425-011-4293-7. Bibcode: 2011ScChA..54.1767S.
External links
- Demailly, Jean-Pierre (June 1996). "L2 estimates for the d-bar operator on complex manifolds, Notes de cours, Ecole d'été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble". https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/estimations_l2.pdf.
- Analytic Methods in Algebraic Geometry (OpenContent book See B5)
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Original source: https://handwiki.org/wiki/Ohsawa–Takegoshi L2 extension theorem
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Original source: https://handwiki.org/wiki/Ohsawa–Takegoshi L2 extension theorem
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