Kähler surfaces with six-positive curvature operator of the second kind
HTML articles powered by AMS MathViewer
- by Xiaolong Li
- Proc. Amer. Math. Soc. 151 (2023), 4909-4922
- DOI: https://doi.org/10.1090/proc/16363
- Published electronically: July 28, 2023
- HTML | PDF | Request permission
Abstract:
The purpose of this article is to initiate the investigation of the curvature operator of the second kind on Kähler manifolds. The main result asserts that a closed Kähler surface with six-positive curvature operator of the second kind is biholomorphic to $\mathbb {CP}^2$. It is also shown that a closed non-flat Kähler surface with six-nonnegative curvature operator of the second kind is either biholomorphic to $\mathbb {CP}^2$ or isometric to $\mathbb {S}^2 \times \mathbb {S}^2$.References
- M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379–392. MR 266084, DOI 10.4310/jdg/1214429060
- Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
- Jean-Pierre Bourguignon and Hermann Karcher, Curvature operators: pinching estimates and geometric examples, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 1, 71–92. MR 493867, DOI 10.24033/asens.1340
- Armand Borel, On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2) 71 (1960), 508–521. MR 111059, DOI 10.2307/1969940
- Simon Brendle, A general convergence result for the Ricci flow in higher dimensions, Duke Math. J. 145 (2008), no. 3, 585–601. MR 2462114, DOI 10.1215/00127094-2008-059
- Simon Brendle, Ricci flow and the sphere theorem, Graduate Studies in Mathematics, vol. 111, American Mathematical Society, Providence, RI, 2010. MR 2583938, DOI 10.1090/gsm/111
- Christoph Böhm and Burkhard Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2) 167 (2008), no. 3, 1079–1097. MR 2415394, DOI 10.4007/annals.2008.167.1079
- Xiaodong Cao, Matthew J. Gursky, and Hung Tran, Curvature of the second kind and a conjecture of Nishikawa, Comment. Math. Helv. 98 (2023), no. 1, 195–216. MR 4592855, DOI 10.4171/cmh/545
- Haiwen Chen, Pointwise $\frac 14$-pinched $4$-manifolds, Ann. Global Anal. Geom. 9 (1991), no. 2, 161–176. MR 1136125, DOI 10.1007/BF00776854
- X. X. Chen, On Kähler manifolds with positive orthogonal bisectional curvature, Adv. Math. 215 (2007), no. 2, 427–445. MR 2355611, DOI 10.1016/j.aim.2006.11.006
- Eugenio Calabi and Edoardo Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 (1960), 472–507. MR 111058, DOI 10.2307/1969939
- Andrzej Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433. MR 707181
- Theodore Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174. MR 123272, DOI 10.2140/pjm.1961.11.165
- S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
- HuiLing Gu and ZhuHong Zhang, An extension of Mok’s theorem on the generalized Frankel conjecture, Sci. China Math. 53 (2010), no. 5, 1253–1264. MR 2653275, DOI 10.1007/s11425-010-0013-y
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
- Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
- Richard S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), no. 1, 1–92. MR 1456308, DOI 10.4310/CAG.1997.v5.n1.a1
- Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435–441. MR 350657
- Toyoko Kashiwada, On the curvature operator of the second kind, Natur. Sci. Rep. Ochanomizu Univ. 44 (1993), no. 2, 69–73. MR 1259209
- Norihito Koiso, A decomposition of the space ${\cal M}$ of Riemannian metrics on a manifold, Osaka Math. J. 16 (1979), no. 2, 423–429. MR 539597
- Norihito Koiso, On the second derivative of the total scalar curvature, Osaka Math. J. 16 (1979), no. 2, 413–421. MR 539596
- Xiaolong Li, Manifolds with nonnegative curvature operator of the second kind, Commun. Contemp. Math. (to appear), arXiv:2112.08465v4, 2021, DOI 10.1142/S021919972350003.
- Xiaolong Li, Manifolds with $4\frac 12$-positive curvature operator of the second kind, J. Geom. Anal. 32 (2022), no. 11, Paper No. 281, 14. MR 4478476, DOI 10.1007/s12220-022-01033-8
- Xiaolong Li, Product manifolds and the curvature operator of the second kind, arXiv:2209.02119, 2022.
- Xiaolong Li, Kähler manifolds and the curvature operator of the second kind, Math. Z. 303 (2023), no. 4, Paper No. 101, 26. MR 4564573, DOI 10.1007/s00209-023-03263-0
- Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
- Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
- Josef Mikeš, Vladimir Rovenski, and Sergey E. Stepanov, An example of Lichnerowicz-type Laplacian, Ann. Global Anal. Geom. 58 (2020), no. 1, 19–34. MR 4117919, DOI 10.1007/s10455-020-09714-9
- Mario J. Micallef and McKenzie Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649–672. MR 1253619, DOI 10.1215/S0012-7094-93-07224-9
- Seiki Nishikawa, On deformation of Riemannian metrics and manifolds with positive curvature operator, Curvature and topology of Riemannian manifolds (Katata, 1985) Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 202–211. MR 859586, DOI 10.1007/BFb0075657
- Jan Nienhaus, Peter Petersen, and Matthias Wink, Betti numbers and the curvature operator of the second kind, arXiv:2206.14218, 2022.
- Koichi Ogiue and Shun-ichi Tachibana, Les variétés riemanniennes dont l’opérateur de courbure restreint est positif sont des sphères d’homologie réelle, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 1, A29–A30 (French, with English summary). MR 545675
- Peter Petersen and Matthias Wink, New curvature conditions for the Bochner technique, Invent. Math. 224 (2021), no. 1, 33–54. MR 4228500, DOI 10.1007/s00222-020-01003-3
- Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, DOI 10.1007/BF01390043
- Shun-ichi Tachibana, A theorem on Riemannian manifolds of positive curvature operator, Proc. Japan Acad. 50 (1974), 301–302. MR 365415
- Burkhard Wilking, A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities, J. Reine Angew. Math. 679 (2013), 223–247. MR 3065160, DOI 10.1515/crelle.2012.018
Bibliographic Information
- Xiaolong Li
- Affiliation: Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, Kansas 67260
- ORCID: 0000-0002-0932-8374
- Email: xiaolong.li@wichita.edu
- Received by editor(s): February 15, 2022
- Received by editor(s) in revised form: November 8, 2022
- Published electronically: July 28, 2023
- Additional Notes: The author’s research was partially supported by Simons Collaboration Grant #962228 and a start-up grant at Wichita State University
- Communicated by: Jiaping Wang
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4909-4922
- MSC (2020): Primary 53C21, 53C55
- DOI: https://doi.org/10.1090/proc/16363
- MathSciNet review: 4634892