On Vanishing Theorems for Locally Conformally Flat Riemannian Manifolds with an Integral Pinching Condition

Abstract

In this paper, we show some vanishing theorems for harmonic p-forms on a locally conformally flat Riemannian manifold. In the concrete, provided that the integral of the traceless Ricci tensor has a suitable bound, we obtain a vanishing theorem for them without any scalar curvature conditions. Another theorem is also given under the condition on nonpositive scalar curvature, which improves and extends the ones previous.

Keywords

Harmonic p-form / Vanishing theorem / Locally conformally flat Riemannian manifold / 58J05 / 58J35

References

[1] Bourguignon JP. Les variétés de dimension 4 a signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math., 1981, 63(2): 263-286

[2] Calderbank DMJ, Gauduchon P, Herzlich M. Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal., 2000, 173(1): 214-255

[3] Chang LC, Guo CL, Anna Sung CJ. p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic 1-forms on complete manifolds. Arch. Math., 2010, 94: 183-192

[4] Chen JTR, Sung CJ. Harmonic forms on manifolds with weighted Poincaré inequality. Pac. J. Math., 2009, 242: 201-214

[5] Dong Y, Lin H, Wei SW. L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} curvature pinching theorems and vanishing theorems on complete Riemannian manifolds. Tohoku Math. J. (2), 2019, 71(4): 581-607

[6] Dung NT, Seo K. p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic functions and connectedness at infinity of complete Riemannian manifolds. Ann. Mat., 2017, 196(4): 1489-1511

[7] Dung NT, Sung CJ. Manifolds with a weighted Poincaré inequality. Proc. Amer. Math. Soc., 2014, 142(5): 1783-1794

[8] Dung NT, Sung CJ. Analysis of weighted p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic forms and applications. Intern. J. Math., 2019, 30(10): 1950058

[9] Goldberg SI. An application of Yau’s maximum principle to conformally flat spaces. Proc. Amer. Math. Soc., 1980, 79: 268-270

[10] Han YB. The topological structure of conformally flat Riemannian manifolds. Results Math., 2018, 73: 54

[11] Han YB, Pan H. Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic 1-forms on the submanifolds in a Hadamard manifold. J. Geom. Phys., 2016, 107: 79-91

[12] Han Y, Zhang Q, Liang M. Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic 1-forms on locally conformally flat Riemannian manifolds. Kodai Math. J., 2017, 40: 518-536

[13] Lam KH. Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc., 2010, 362(10): 5043-5062

[14] Lin HZ. On the structure of conformally flat Riemannian manifolds. Nonlinear Anal., 2015, 123–124: 115-125

[15] Lin HZ. Vanishing theorem for complete Riemannian manifolds with nonnegative scalar curvature. Geom. Dedicata., 2019, 201: 187-201

[16] Li P, Wang JP. Weighted Poincaré inequality and rigidity of complete manifolds. Ann. Sci. Éc. Norm. Sup., 2016, 39: 921-982

[17] Nguyen DT, Pham DT. On vanishing theorems for locally conformally flat Riemannian manifolds. Bull. Korean Math. Soc., 2022, 59(2): 469-479

[18] Vieira M. Vanishing theorems for L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} harmonic forms on complete Riemannian manifolds. Geom. Dedicata., 2016, 184: 175-191

[19] Zhang X. A note on p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-harmonic 1-forms on complete manifolds. Canad. Math. Bull., 2011, 44: 376-384

[20] Zhou J. Vanishing theorems for L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document} harmonic p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-forms on Riemannian manifolds with a weighted p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}-Poincaré inequality. J. Math. Anal. Appl., 2020, 490: 124229

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature