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Analytic and geometric properties of generic Ricci solitons


Authors: G. Catino, P. Mastrolia, D. D. Monticelli and M. Rigoli
Journal: Trans. Amer. Math. Soc. 368 (2016), 7533-7549
MSC (2010): Primary 53C20, 53C25
DOI: https://doi.org/10.1090/tran/6864
Published electronically: March 3, 2016
MathSciNet review: 3546774
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Abstract: The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three-dimensional generic shrinking Ricci soliton is given by quotients of either $ \mathbb{S}^3$, $ \mathbb{R}\times \mathbb{S}^2$ or $ \mathbb{R}^3$ under some very weak conditions on the vector field $ X$ generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the $ X$-Laplacian on every generic Ricci soliton without any assumption on $ X$.


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Additional Information

G. Catino
Affiliation: Department of Mathematics, Politecnico di Milano, 20133 Milan, Italy
Email: giovanni.catino@polimi.it

P. Mastrolia
Affiliation: Department of Mathematics, Università degli Studi di Milano, 20133 Milan, Italy
Email: paolo.mastrolia@gmail.com

D. D. Monticelli
Affiliation: Department of Mathematics, Università degli Studi di Milano, 20133 Milan, Italy
Address at time of publication: Department of Mathematics, Politecnico di Milano, 20133 Milan, Italy
Email: dario.monticelli@polimi.it

M. Rigoli
Affiliation: Department of Mathematics, Università degli Studi di Milano, 20133 Milan, Italy
Email: marco.rigoli@unimi.it

DOI: https://doi.org/10.1090/tran/6864
Keywords: Ricci solitons, Omori-Yau maximum principle, rigidity results
Received by editor(s): May 24, 2014
Published electronically: March 3, 2016
Additional Notes: The first author was supported by GNAMPA projects “Equazioni differenziali con invarianze in analisi globale” and “Equazioni di evoluzione geometriche e strutture di tipo Einstein”
The second author was partially supported by FSE, Regione Lombardia
The third author was supported by GNAMPA projects “Equazioni differenziali con invarianze in analisi globale” and “Analisi Globale ed Operatori Degeneri”.
The first, second and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
Article copyright: © Copyright 2016 American Mathematical Society