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A Weitzenböck formula for canonical metrics on four-manifolds


Author: Peng Wu
Journal: Trans. Amer. Math. Soc. 369 (2017), 1079-1096
MSC (2010): Primary 53C25; Secondary 53C24
DOI: https://doi.org/10.1090/tran/6964
Published electronically: July 26, 2016
MathSciNet review: 3572265
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Abstract: We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds. As applications, we classify Einstein four-manifolds and conformally Einstein four-manifolds with half two-nonnegative curvature operator, which in some sense provides a characterization of Kähler-Einstein metrics and Hermitian, Einstein metrics with positive scalar curvature on four-manifolds, respectively. We also discuss the classification of four-dimensional gradient shrinking Ricci solitons with half two-nonnegative curvature operator and half harmonic Weyl curvature.


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  • [1] D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206 (French). MR 889476, https://doi.org/10.1007/BFb0075847
  • [2] Marcel Berger, Sur quelques variétés d'Einstein compactes, Ann. Mat. Pura Appl. (4) 53 (1961), 89-95 (French). MR 0130659
  • [3] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • [4] Simon Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151 (2010), no. 1, 1-21. MR 2573825, https://doi.org/10.1215/00127094-2009-061
  • [5] Huai-Dong Cao and Qiang Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149-1169. MR 3053567, https://doi.org/10.1215/00127094-2147649
  • [6] Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu, Recent developments on Hamilton's Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47-112. MR 2488948, https://doi.org/10.4310/SDG.2007.v12.n1.a3
  • [7] Xiaodong Cao, Biao Wang, and Zhou Zhang, On locally conformally flat gradient shrinking Ricci solitons, Commun. Contemp. Math. 13 (2011), no. 2, 269-282. MR 2794486, https://doi.org/10.1142/S0219199711004191
  • [8] Jeffrey S. Case, Smooth metric measure spaces and quasi-Einstein metrics, Internat. J. Math. 23 (2012), no. 10, 1250110, 36. MR 2999055, https://doi.org/10.1142/S0129167X12501108
  • [9] Jeffrey S. Case, The energy of a smooth metric measure space and applications, J. Geom. Anal. 25 (2015), no. 1, 616-667. MR 3299295, https://doi.org/10.1007/s12220-013-9441-6
  • [10] Jeffrey S. Case, A Yamabe-type problem on smooth metric measure spaces, J. Differential Geom. 101 (2015), no. 3, 467-505. MR 3415769
  • [11] Jeffrey Case, Yu-Jen Shu, and Guofang Wei, Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), no. 1, 93-100. MR 2784291, https://doi.org/10.1016/j.difgeo.2010.11.003
  • [12] Giovanni Catino, A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds, Differential Geom. Appl. 30 (2012), no. 6, 660-664. MR 2996860, https://doi.org/10.1016/j.difgeo.2012.09.005
  • [13] Giovanni Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), no. 3-4, 751-756. MR 2945582, https://doi.org/10.1007/s00209-011-0888-5
  • [14] Giovanni Catino and Carlo Mantegazza, The evolution of the Weyl tensor under the Ricci flow, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1407-1435 (2012) (English, with English and French summaries). MR 2951497, https://doi.org/10.5802/aif.2644
  • [15] Giovanni Catino, P. Mastrolia, D. Monticelli, and M. Rigoli, On the geometry of gradient Einstein-type manifolds, arXiv:math.DG/1402.3453, 2014.
  • [16] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, A conformally invariant sphere theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 105-143. MR 2031200, https://doi.org/10.1007/s10240-003-0017-z
  • [17] Xiuxiong Chen, Claude Lebrun, and Brian Weber, On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 1137-1168. MR 2425183, https://doi.org/10.1090/S0894-0347-08-00594-8
  • [18] Xiuxiong Chen and Yuanqi Wang, On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal. 25 (2015), no. 2, 1335-1343. MR 3319974, https://doi.org/10.1007/s12220-014-9471-8
  • [19] Andrzej Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405-433. MR 707181
  • [20] Manolo Eminenti, Gabriele La Nave, and Carlo Mantegazza, Ricci solitons: the equation point of view, Manuscripta Math. 127 (2008), no. 3, 345-367. MR 2448435, https://doi.org/10.1007/s00229-008-0210-y
  • [21] Michael Feldman, Tom Ilmanen, and Lei Ni, Entropy and reduced distance for Ricci expanders, J. Geom. Anal. 15 (2005), no. 1, 49-62. MR 2132265, https://doi.org/10.1007/BF02921858
  • [22] Manuel Fernández-López and Eduardo García-Río, Rigidity of shrinking Ricci solitons, Math. Z. 269 (2011), no. 1-2, 461-466. MR 2836079, https://doi.org/10.1007/s00209-010-0745-y
  • [23] Joel Fine, Kirill Krasnov, and Dmitri Panov, A gauge theoretic approach to Einstein 4-manifolds, New York J. Math. 20 (2014), 293-323. MR 3193955
  • [24] A. Rod Gover and Paweł Nurowski, Obstructions to conformally Einstein metrics in $ n$ dimensions, J. Geom. Phys. 56 (2006), no. 3, 450-484. MR 2171895, https://doi.org/10.1016/j.geomphys.2005.03.001
  • [25] Matthew J. Gursky, Four-manifolds with $ \delta W^+=0$ and Einstein constants of the sphere, Math. Ann. 318 (2000), no. 3, 417-431. MR 1800764, https://doi.org/10.1007/s002080000130
  • [26] Matthew J. Gursky and Claude Lebrun, On Einstein manifolds of positive sectional curvature, Ann. Global Anal. Geom. 17 (1999), no. 4, 315-328. MR 1705915, https://doi.org/10.1023/A:1006597912184
  • [27] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497
  • [28] Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237-262. MR 954419, https://doi.org/10.1090/conm/071/954419
  • [29] Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47-62. MR 806701
  • [30] Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301-307. MR 1249376, https://doi.org/10.1016/0926-2245(93)90008-O
  • [31] Carlos N. Kozameh, Ezra T. Newman, and K. P. Tod, Conformal Einstein spaces, Gen. Relativity Gravitation 17 (1985), no. 4, 343-352. MR 788800, https://doi.org/10.1007/BF00759678
  • [32] Claude LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277-302. MR 2899877
  • [33] Claude LeBrun, Einstein metrics, harmonic forms, and symplectic four-manifolds, Ann. Global Anal. Geom. 48 (2015), no. 1, 75-85. MR 3351078, https://doi.org/10.1007/s10455-015-9458-0
  • [34] Pengzi Miao and Luen-Fai Tam, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 141-171. MR 2546025, https://doi.org/10.1007/s00526-008-0221-2
  • [35] Pengzi Miao and Luen-Fai Tam, Einstein and conformally flat critical metrics of the volume functional, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2907-2937. MR 2775792, https://doi.org/10.1090/S0002-9947-2011-05195-0
  • [36] Ovidiu Munteanu and Natasa Sesum, On gradient Ricci solitons, J. Geom. Anal. 23 (2013), no. 2, 539-561. MR 3023848, https://doi.org/10.1007/s12220-011-9252-6
  • [37] Lei Ni and Nolan Wallach, On a classification of gradient shrinking solitons, Math. Res. Lett. 15 (2008), no. 5, 941-955. MR 2443993, https://doi.org/10.4310/MRL.2008.v15.n5.a9
  • [38] D. Page, A compact rotating gravitational instanton, Phys. Lett. 79 B (1979), 235-238.
  • [39] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
  • [40] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.
  • [41] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245.
  • [42] Peter Petersen and William Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010), no. 4, 2277-2300. MR 2740647, https://doi.org/10.2140/gt.2010.14.2277
  • [43] Thomas Richard and Harish Seshadri, Positive isotropic curvature and self-duality in dimension 4, Manuscripta Math. 149 (2016), no. 3-4, 443-457. MR 3458177, https://doi.org/10.1007/s00229-015-0790-2
  • [44] I. M. Singer and J. A. Thorpe, The curvature of $ 4$-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355-365. MR 0256303
  • [45] J.-Y. Wu, P. Wu, and W. Wylie, Gradient shrinking Ricci solitons of half harmonic Weyl curvature, arXiv:math.DG/1410.7303.
  • [46] P. Wu, Studies on Einstein manifolds and Ricci solitons, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)-University of California, Santa Barbara. MR 3094048
  • [47] P. Wu, A note on Einstein four-manifolds with positive sectional curvature, preprint, 2012.
  • [48] P. Wu, Einstein four-manifolds with three-positive curvature operator are half conformally flat, preprint, 2013.
  • [49] P. Wu, Rigidity of quasi-Einstein four-manifolds with half harmonic Weyl curvature, preprint, 2015.
  • [50] DaGang Yang, Rigidity of Einstein $ 4$-manifolds with positive curvature, Invent. Math. 142 (2000), no. 2, 435-450. MR 1794068, https://doi.org/10.1007/PL00005792
  • [51] Zhu-Hong Zhang, Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math. 242 (2009), no. 1, 189-200. MR 2525510, https://doi.org/10.2140/pjm.2009.242.189

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Additional Information

Peng Wu
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Address at time of publication: Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: wupenguin@fudan.edu.cn

DOI: https://doi.org/10.1090/tran/6964
Keywords: Weitzenb\"ock formula, Berger curvature decomposition, half two-positive curvature operator, half positive isotropic curvature, gradient Ricci soliton, quasi-Einstein manifold, conformally Einstein manifold, generalized quasi-Einstein manifolds, canonical metric, smooth metric measure space.
Received by editor(s): February 9, 2015
Published electronically: July 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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