Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms
In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then ( M , g ) is itself a complex sp...
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| Vydané v: | The Journal of geometric analysis Ročník 31; číslo 8; s. 7842 - 7865 |
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| Jazyk: | English |
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New York
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01.08.2021
Springer Nature B.V |
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| ISSN: | 1050-6926, 1559-002X |
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| Abstract | In this paper, we address the problem of studying those complex manifolds
M
equipped with extremal metrics
g
induced by finite or infinite-dimensional complex space forms. We prove that when
g
is assumed to be radial and the ambient space is finite-dimensional, then (
M
,
g
) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric
g
has constant scalar curvature and is
well behaved
(see Definition
1
in the Introduction). Finally, we analyze the radial Kähler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature. |
|---|---|
| AbstractList | In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then (M, g) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kähler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature. In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then ( M , g ) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kähler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature. |
| Author | Salis, Filippo Zuddas, Fabio Loi, Andrea |
| Author_xml | – sequence: 1 givenname: Andrea orcidid: 0000-0003-0407-7403 surname: Loi fullname: Loi, Andrea email: loi@unica.it organization: Dipartimento di Matematica, Università di Cagliari – sequence: 2 givenname: Filippo surname: Salis fullname: Salis, Filippo organization: Istituto Nazionale di Alta Matematica, Politecnico di Torino – sequence: 3 givenname: Fabio surname: Zuddas fullname: Zuddas, Fabio organization: Dipartimento di Matematica, Università di Cagliari |
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| Cites_doi | 10.1142/S0129167X04002429 10.1007/s00013-017-1055-y 10.1016/j.geomphys.2018.07.013 10.2969/jmsj/02710076 10.1016/j.crma.2004.11.028 10.1512/iumj.2004.53.2311 10.1016/j.jmaa.2014.09.080 10.1090/gsm/152 10.4099/math1924.4.171 10.1215/00127094-1593308 10.1007/BF01404460 10.24033/bsmf.2281 10.1007/s00222-014-0543-y 10.1007/s00208-002-0349-x 10.1007/BF01430959 10.1007/s10711-015-0085-5 10.1090/S0002-9904-1947-08778-4 10.1007/s00209-017-2033-6 10.2307/1969817 10.4310/AJM.2012.v16.n3.a7 10.2748/tmj/1178228285 10.1215/00127094-2011-001 10.1007/s00208-010-0554-y 10.1007/s10455-019-09680-x 10.1007/BF01457231 10.1007/BF02921947 10.1016/j.jmaa.2019.123715 10.1007/978-3-319-99483-3 |
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Phys. – volume: 290 start-page: 599 year: 2018 ident: 554_CR26 publication-title: Math. Z. doi: 10.1007/s00209-017-2033-6 – volume: 340 start-page: 287 issue: 4 year: 2005 ident: 554_CR9 publication-title: C. R. Math. Acad. Sci. Paris doi: 10.1016/j.crma.2004.11.028 – volume: 133 start-page: 210 year: 2018 ident: 554_CR27 publication-title: J. Geom. Phys. doi: 10.1016/j.geomphys.2018.07.013 – volume: 56 start-page: 583 issue: 3 year: 2019 ident: 554_CR20 publication-title: Ann. Glob. Anal. Geom. doi: 10.1007/s10455-019-09680-x – volume: 114 start-page: 171 issue: 1 year: 2020 ident: 554_CR34 publication-title: J. Diff. Geom. – volume: 1 start-page: 21 year: 1967 ident: 554_CR10 publication-title: J. Diff. Geom. – volume: 16 start-page: 479 issue: 3 year: 2012 ident: 554_CR11 publication-title: Asian J. Math. doi: 10.4310/AJM.2012.v16.n3.a7 – volume: 161 start-page: 1411 issue: 8 year: 2012 ident: 554_CR31 publication-title: Duke Math. J. doi: 10.1215/00127094-1593308 – volume: 53 start-page: 179 year: 1947 ident: 554_CR3 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1947-08778-4 – volume-title: An Introduction to Extremal Kähler metrics year: 2014 ident: 554_CR32 doi: 10.1090/gsm/152 – volume: 200 start-page: 925 issue: 3 year: 2015 ident: 554_CR33 publication-title: Invent. Math. doi: 10.1007/s00222-014-0543-y – volume: 15 start-page: 531 year: 2004 ident: 554_CR29 publication-title: Int. J. Math. doi: 10.1142/S0129167X04002429 – volume: 484 start-page: 123715 issue: 1 year: 2020 ident: 554_CR28 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2019.123715 – volume: 157 start-page: 1 issue: 1 year: 2011 ident: 554_CR2 publication-title: Duke Math. J. doi: 10.1215/00127094-2011-001 – volume: 350 start-page: 145 year: 2011 ident: 554_CR23 publication-title: Math. Ann. doi: 10.1007/s00208-010-0554-y – volume: 1 start-page: 369 year: 1967 ident: 554_CR17 publication-title: J. Diff. Geom. – volume-title: Kähler Immersions of Kähler Manifolds into Complex Space Forms year: 2018 ident: 554_CR24 doi: 10.1007/978-3-319-99483-3 – volume: 102 start-page: 259 year: 1982 ident: 554_CR6 publication-title: Ann. Math. Stud. |
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| Snippet | In this paper, we address the problem of studying those complex manifolds
M
equipped with extremal metrics
g
induced by finite or infinite-dimensional complex... In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex... |
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| SubjectTerms | Abstract Harmonic Analysis Convex and Discrete Geometry Curvature Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics |
| Title | Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms |
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| Volume | 31 |
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