The New Gap Theorem for Certain Riemannian Manifolds

Abstract

In this paper, the authors investigate the geometric rigidity of Riemannian manifolds under suitable curvature restrictions. The authors first prove a new gap theorem for the Ricci curvature of compact locally conformally flat Riemannian manifolds. Subsequently, the authors consider the Riemannian manifolds with the Cotton tensor C satisfying div C = 0 and prove some integral curvature pinching theorems.

Keywords

Gap theorems / Locally conformally flat manifolds / Ricci curvature / Constant scalar curvature / Cotton tensor / 53C20 / 53C25

References

[1] Aubin T. The scalar curvature, Differential Geometry and Relativity, 1976, Dordrecht-Boston, Mass., Reidel518 3

[2] Bour V, Carron G. Optimal integral pinching results. Ann. Sci. Éc. Norm. Supér., 2015, 48(1): 41-70

[3] Carron G. Some old and new results about rigidity of critical metric. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2014, 13(4): 1091-1113

[4] Carron G, Herzlich M. The Huber theorem for non-compact conformally flat manifolds. Comment. Math. Helv., 2002, 77(1): 192-220

[5] Catino G. On conformally flat manifolds with constant positive scalar curvature. Proc. Amer. Math. Soc., 2016, 144(6): 2627-2634

[6] Chang A, Gursky M, Yang P C. A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes études Sci., 2003, 98: 105-143

[7] Chang A, Qing J, Yang P C. On a conformal gap and finiteness theorem for a class of four-manifolds. Geom. Funct. Anal., 2007, 17(2): 404-434

[8] Chang S P. A closed hypersurface with constant scalar and mean curvatures in S4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb S}^{4}$$\end{document} is isoparametric. Comm. Anal. Geom., 1993, 1(1): 71-100

[9] Chang S P. On closed hypersurfaces of constant scalar curvatures and mean curvatures in Sn+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb S}^{n+1}$$\end{document}. Pacific J. Math., 1994, 165(1): 67-76

[10] Cheng Q M. Compact locally conformally flat Riemannian manifolds. Bull. London Math. Soc., 2001, 33(4): 459-465

[11] Cheng Q M, Ishikawa S, Shiohama K. Conformally flat 3-manifolds with constant scalar curvature. J. Math. Soc. Japan, 1999, 51(1): 209-226

[12] Cheng X X, Hu Z J. On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature - submanifolds versions. J. Math. Anal. Appl., 2018, 464(2): 1147-1157

[13] de Almeida S C, Brito F G B. Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature. Duke Math. J., 1990, 61(1): 195-206

[14] DeTurck D M, Goldschmidt H. Regularity theorems in Riemannian geometry, II, Harmonic curvature and the Weyl tensor. Forum Math., 1989, 1(4): 377-394

[15] Ding Q, Xin Y L. On Chern’s problem for rigidity of minimal hypersurfaces in the spheres. Adv. Math., 2011, 227(1): 131-145

[16] Dong Y X, Lin H Z, Wei S W. L² curvature pinching theorems and vanishing theorems on complete Riemannian manifolds. Tohoku Math. J., 2019, 71(4): 581-607

[17] Eisenhart L P. Riemannian Geometry, 1997, Princeton, Princeton University PressEighth printing

[18] Fu H P. On compact manifolds with harmonic curvature and positive scalar curvature. J. Geom. Anal., 2017, 27(4): 3120-3139

[19] Fu H P, Peng J K. Conformally flat Riemannian manifolds with finite L^p-norm curvature. Ann. Mat. Pura Appl., 2017, 196(4): 1903-1912

[20] Fu H P, Xiao L Q. Some L^p rigidity results for complete manifolds with harmonic curvature. Potential Anal., 2018, 48(2): 239-255

[21] Goldberg S I. An application of Yau’s maximum principle to conformally flat spaces. Proc. Amer. Math. Soc., 1980, 79(2): 268-270

[22] Goldberg S I, Okumura M. Conformally flat manifolds and a pinching problem on the Ricci tensor. Proc. Amer. Math. Soc., 1976, 58(1): 234-236

[23] Gu J R, Lei L, Xu H W. A new gap for complete hypersurfaces with constant mean curvature in space forms. Pacific J. Math., 2022, 318(1): 51-67

[24] Hasanis T. Conformally flat spaces and a pinching problem on the Ricci tensor. Proc. Amer. Math. Soc., 1996, 86(2): 312-315

[25] Hebey E, Vaugon M. Effective L^p pinching for the concircular curvature. J. Geom. Anal., 1996, 6(4): 531-553

[26] Huisken G. Ricci deformation of the metric on a Riemannian manifold. J. Differential Geom., 1985, 21(1): 47-62

[27] Lee J M, Parker T H. The Yamabe problem. Bull. Amer. Math. Soc., 1987, 17(1): 37-91

[28] Lei L, Xu H W, Xu Z Y. On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere. Sci. China Math., 2021, 64(7): 1493-1504

[29] Lei L, Xu H W, Xu Z Y. An optimal gap theorem for scalar curvature of CMC hypersurfaces in a sphere. Proc. Amer. Math. Soc., 2023, 151(2): 811-821

[30] Lin H Z. L^p-vanishing results for conformally flat manifolds and submanifolds. J. Geom. Phys., 2013, 73: 157-165

[31] Mastrolia P, Monticelli D, Rigoli M. A note on curvature of Riemannian manifolds. J. Math. Anal. Appl., 2013, 399(2): 505-513

[32] Peng C K, Terng C L. Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on Minimal Submanifolds, 1983, Princeton, Princeton Univ. Press177198103

[33] Peng C K, Terng C L. The scalar curvature of minimal hypersurfaces in spheres. Math. Ann., 1983, 266: 105-113

[34] Pigola S, Rigoli M, Setti A G. Some characterizations of space-forms. Trans. Amer. Math. Soc., 2007, 359(4): 1817-1828

[35] Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 1984, 20(2): 479-495

[36] Shen Z M. Some rigidity phenomena for Einstein metrics. Proc. Amer. Math. Soc., 1990, 108(4): 981-987

[37] Shen Z M. Rigidity theorems for nonpositive Einstein metrics. Proc. Amer. Math. Soc., 1992, 116(4): 1107-1114

[38] Singer M A. Positive Einstein metrics with small Ln2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{L}^{n \over 2}}$$\end{document}-norm of the Weyl tensor. Differential Geom. Appl., 1992, 2(3): 269-274

[39] Tani M. On a conformally flat Riemannian space with positive Ricci curvature. Tohoku Math. J., 1967, 19(2): 227-231

[40] Trudinger N S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1968, 22(3): 265-274

[41] Xin Y L. Minimal Submanifolds and Related Topics, 2019Second editionHackensack, NJ, World Scientific Publishing Co. Pte. Ltd.16

[42] Xu H W, Tian L. A new pinching theorem for closed hypersurfaces with constant mean curvature in Sn+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb S}^{n+1}$$\end{document}. Asian J. Math., 2011, 15(4): 611-630

[43] Xu H W, Xu Z Y. The second pinching theorem for hypersurfaces with constant mean curvature in a sphere. Math. Ann., 2013, 356(3): 869-883

[44] Xu H W, Xu Z Y. A new characterization of the Clifford torus via scalar curvature pinching. J. Funct. Anal., 2014, 267(10): 3931-3962

[45] Xu H W, Zhao E T. L^p Ricci curvature pinching theorems for conformally flat Riemannian manifolds. Pacific J. Math., 2010, 245(2): 381-396

[46] Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka J. Math., 1960, 12(1): 21-37

[47] Yang H C, Cheng Q M. Chern’s conjecture on minimal hypersurfaces. Math. Z., 1998, 227(3): 377-395

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg