Document Zbl 1454.53059

Hermitian metrics of constant Chern scalar curvature on ruled surfaces. (English) Zbl 1454.53059

Kodai Math. J. 43, No. 3, 409-430 (2020).

Summary: It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric [V. Apostolov et al., Adv. Math. 227, No. 6, 2385–2424 (2011; Zbl 1232.32011); P. Gauduchon, “Calabi’s extremal Kähler metrics: an elementary introduction”, Preprint; J. Martinez-Garcia, “Constant scalar curvature Kähler metrics on rational surfaces”, Preprint, arXiv:1712.04857]. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page-Bérard-Bergery’s ansatz [D. Page, “A compact rotating gravitational instanton”, Phys. Lett. B 79, No. 3, 235–238 (1978; doi:10.1016/0370-2693(78)90231-9); L. Bérard Bergery, Inst. Élie Cartan, Univ. Nancy I 6, 1–60 (1983; Zbl 0544.53038) We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by P. Gauduchon [C. R. Acad. Sci., Paris, Sér. A 290, 327–330 (1980; Zbl 0422.58015); Math. Ann. 267, 495–518 (1984; Zbl 0523.53059)].

Cited in 1 Review

Cited in 4 Documents

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53B35	Local differential geometry of Hermitian and Kählerian structures

Keywords:

Hermitian metrics; Chern scalar curvature; ruled surfaces; Hirzebruch surfaces; Kähler metrics

Citations:

Zbl 1232.32011; Zbl 0544.53038; Zbl 0422.58015; Zbl 0523.53059

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Full Text: DOI arXiv Euclid

References:

[1]	T. Aubin,E´ quations di¤e´rentielles non line´aires et proble‘me de Yamabe concernant la courbure scalaire,J. Math. Pures Appl.(9) 55(1976), 269-296. · Zbl 0336.53033
[2]	T.Aubin,Some nonlinear problems in Riemannian geometry,Springer monographs in mathematics, Springer-Verlag, Berlin, 1998. · Zbl 0896.53003
[3]	A. Bahri and H. Brezis,E´ quations elliptiques non line´aires sur des varie´te´s avec exposant de Sobolev critique,C. R. Acad. Sci. Paris Se´r. I Math.307(1988), 573-576. · Zbl 0694.35059
[4]	A. Balas,Compact Hermitian manifolds of constant holomorphic sectional curvature,Math. Z.189(1985), 193-210. · Zbl 0575.53044
[5]	L.Be´ rard-Bergery,Sur de nouvelles varie´te´s riemanniennes d’Einstein,In Institut E´ lie Cartan6, Univ. Nancy, Nancy, 1982, 1-60. · Zbl 0544.53038
[6]	M. S. Berger,On Hermitian structures of prescribed nonpositive Hermitian scalar curvature, Bull. Amer. Math. Soc.78(1972), 734-736. · Zbl 0259.53051
[7]	A.L.Besse,Einstein manifolds,Classics in mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition.
[8]	H. Brezis,Points critiques dans les proble‘mes variationnels sans compacite´,Aste´risque, 161- 162 (1988), Exp. No.698,5, 239-256 (1989), Se´minaire Bourbaki, Vol. 1987/88. · Zbl 0860.58007
[9]	S.CalamaiandF.Zou,A note on Chern-Yamabe problem,arXiv preprint 1904.03831 (2019).
[10]	A.J.DiScalaandL.Vezzoni,Chern-ﬂat and Ricci-ﬂat invariant almost Hermitian structures,Ann. Global Anal. Geom.40(2011), 21-45. · Zbl 1218.53032
[11]	A. J. Di Scala and L. Vezzoni,Quasi-Ka¨hler manifolds with trivial Chern holonomy,Math. Z.271(2012), 95-108. · Zbl 1264.53037
[12]	G. Ganchev and S. Ivanov,Compact Hermitian surfaces of Einstein type with respect to the Hermitian connection,Monatsh. Math.123(1997), 53-59. · Zbl 0953.53045
[13]	P. Gauduchon,Calabi’s extremal Ka¨hler metrics: An elementary introduction,In preparation.
[14]	P. Gauduchon,Fibre´s hermitiens a‘ endomorphisme de Ricci non ne´gatif,Bull. Soc. Math. France105(1977), 113-140. · Zbl 0382.53045
[15]	P.Gauduchon,Le the´ore‘me de l’excentricite´ nulle,C. R. Acad. Sci. Paris Se´r. A-B285 (1977), A387-A390. · Zbl 0362.53024
[16]	P. Gauduchon,La topologie d’une surface hermitienne d’Einstein,C. R. Acad. Sci. Paris Se´r. A-B290(1980), A509-A512. · Zbl 0435.53036
[17]	P. Gauduchon,Variation des courbures scalaires en ge´ome´trie hermitienne,C. R. Acad. Sci. Paris Se´r. A-B290(1980), A327-A330. · Zbl 0422.58015
[18]	P. Gauduchon,La 1-forme de torsion d’une varie´te´ hermitienne compacte,Math. Ann.267 (1984), 495-518. · Zbl 0523.53059
[19]	P. Gauduchon,Hermitian connections and Dirac operators,Boll. Un. Mat. Ital. B(7) 11,2, suppl. (1997), 257-288. · Zbl 0876.53015
[20]	P.GauduchonandS.Ivanov,Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4,Math. Z.226(1997), 317-326. · Zbl 1006.53061
[21]	P. T. Ho,Results related to the Chern-Yamabe ﬂow,J Geom Anal. (2019), 1-34.
[22]	C. Koca and C. W. TØnnesen-Friedman,Strongly Hermitian Einstein-Maxwell solutions on ruled surfaces,Ann. Global Anal. Geom.50(2016), 29-46. · Zbl 1347.53057
[23]	T. Koda,A remark on the manifoldCP2aCP2with Be´rard-Bergery’s metric,Ann. Global Anal. Geom.11(1993), 323-329. · Zbl 0816.53029
[24]	M. Lejmi and A. Maalaoui,On the Chern-Yamabe ﬂow,J. Geom. Anal.28(2018), 2692- 2706. · Zbl 1462.53012
[25]	M.LejmiandM.Upmeier,Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry,To appear in Communications in Analysis and Geometry, arXiv:1703.01323 (2017).
[26]	P.Libermann,Sur les connexions hermitiennes,C. R. Acad. Sci. Paris239(1954), 1579- 1581. · Zbl 0057.14203
[27]	A.Lichnerowicz,The´orie globale des connexions et des groupes d’holonomie,Consiglio nazionale delle ricerche: Monograﬁe matematiche2, Edizioni Cremonese, Rome; Dunod, Paris, 1962.
[28]	K. Liu and X. Yang,Ricci curvatures on Hermitian manifolds,Trans. Amer. Math. Soc. 369(2017), 5157-5196. · Zbl 1365.53065
[29]	J.Martinez-Garcia,Constant scalarcurvature Ka¨hler metrics on rational surfaces, arXiv:1712.04857 (2017).
[30]	A. Newlander andL. Nirenberg,Complex analytic coordinates in almost complex manifolds,Ann. of Math.(2) 65(1957), 391-404. · Zbl 0079.16102
[31]	N.Otoba,Constant scalar curvature metrics on Hirzebruch surfaces,Ann. Global Anal. Geom.46(2014), 197-223. · Zbl 1302.53039
[32]	D. N. Page,A compact rotating gravitational instanton,Physics Letters B79(1978), 235- 238.
[33]	X.S.Shen,Estimates for metrics of constant Chern scalar curvature,arXiv:1909.13445 (2019).
[34]	A. Swann and H. Pedersen,Einstein-weyl geometry, the Bach tensor and conformal scalar curvature,Crelle’s Journal441(1993), 99-113. · Zbl 0776.53027
[35]	L. Vezzoni,A note on canonical Ricci forms on 2-step nilmanifolds,Proc. Amer. Math. Soc. 141(2013), 325-333. · Zbl 1272.53020
[36]	H.

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