Hostname: page-component-7f64f4797f-8cpv7 Total loading time: 0 Render date: 2025-11-07T20:36:03.821Z Has data issue: false hasContentIssue false
  • English
  • Français

Structure of generalized Yamabe solitons and its applications

Part of: Global differential geometry

Published online by Cambridge University Press:  07 March 2024

Shun Maeta Open the ORCID record for Shun Maeta [Opens in a new window]
Shun Maeta*
Affiliation:
Department of Mathematics, Faculty of Education, Chiba University, Chiba-shi, Chiba, Japan Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University, Chiba-shi, Chiba, Japan (shun.maeta@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman’s conjecture.

Keywords

Yamabe solitonsYamabe flowk-Yamabe solitonsconformal gradient solitons

MSC classification

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C20: Global Riemannian geometry, including pinching

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 2 , May 2024 , pp. 336 - 348
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Cunha, A. W. and Lima, E. L., A note on gradient k-Yamabe solitons, Ann. Global Anal. Geom. 62(1) (2022), 73–82.CrossRefGoogle Scholar
Bianchi, D. and Setti, A. G., Laplacian cut-offs, porous and fast diffusion on manifolds and other applications, Calc. Var. 57 (2018), .CrossRefGoogle Scholar
Barbosa, E. and Ribeiro, E., On conformal solutions of the Yamabe flow, Arch. Math. 101 (2013), 79–89.CrossRefGoogle Scholar
Bo, L., Ho, P. T. and Sheng, W., The k-Yamabe solitons and the quotient Yamabe solitons, Nonl. Anal. 166 (2018), 181–195.CrossRefGoogle Scholar
Caminha, A., Sousa, P. and Camargo, F., Complete foliations of space forms by hypersurfaces, Bull. Braz. Math. Soc. 41 (2010), 339–353.CrossRefGoogle Scholar
Cao, H.-D., Sun, X. and Zhang, Y., On the structure of gradient Yamabe solitons, Math. Res. Lett. 19 (2012), 767–774.CrossRefGoogle Scholar
Catino, G., Mantegazza, C. and Mazzieri, L., On the global structure of conformal gradient solitons with nonnegative Ricci tensor, Commun. Contemp. Math. 14 (2012), .CrossRefGoogle Scholar
Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math. 144 (1996), 189–237.CrossRefGoogle Scholar
Daskalopoulos, P. and Sesum, N., The classification of locally conformally flat Yamabe solitons, Adv. Math. 240 (2013), 346–369.CrossRefGoogle Scholar
Farina, A., Mari, L. and Valdinoci, E., Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Commun. Partial. Differ. Equ. 38 (2013), 1818–1862.CrossRefGoogle Scholar
Fujii, S. and Maeta, S., Classification of generalized Yamabe solitons in Euclidean spaces, Int. J. Math. 32 (2021), .CrossRefGoogle Scholar
Grigor’yan, A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. math. Soc. 36 (1999), 135–249.CrossRefGoogle Scholar
Hamilton, R., Lectures on geometric flows (1989), unpublished.Google Scholar
Ma, L. and Miquel, V., Remarks on scalar curvature of Yamabe solitons, Ann. Glob. Anal. Geom. 42 (2012), 195–205.CrossRefGoogle Scholar
Maeta, S., Classification of gradient conformal solitons, arXiv:2107.05487[math DG].Google Scholar
Petersen, P., Riemannian Geometry, 3rd edn., Vol. 171, Graduate Texts in Mathematics (New York, NY: Springer, 2016).CrossRefGoogle Scholar
Schoen, R. and Yau, S. T., Lectures on differential geometry, In: Conference Proceedings and Lecture Notes in Geometry and Topology (International Press, Cambridge, 1994).Google Scholar
Tashiro, Y., Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275.CrossRefGoogle Scholar
Yau, S. T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar
Zeng, F., On the h-almost Yamabe soliton, J. Math. Study 54 (2021), 371–386.Google Scholar