Analytic and geometric properties of generic Ricci solitons
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- by G. Catino, P. Mastrolia, D. D. Monticelli and M. Rigoli PDF
- Trans. Amer. Math. Soc. 368 (2016), 7533-7549 Request permission
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 $X$.References
- Shmuel Agmon, Bounds on exponential decay of eigenfunctions of Schrödinger operators, Schrödinger operators (Como, 1984) Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 1–38. MR 824986, DOI 10.1007/BFb0080331
- L. J. Alías, M. Dajczer, and M. Rigoli, Higher order mean curvature estimates for bounded complete hypersurfaces, Nonlinear Anal. 84 (2013), 73–83. MR 3034572, DOI 10.1016/j.na.2013.01.016
- Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112. MR 2488948, DOI 10.4310/SDG.2007.v12.n1.a3
- Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
- Giovanni Catino, Complete gradient shrinking Ricci solitons with pinched curvature, Math. Ann. 355 (2013), no. 2, 629–635. MR 3010141, DOI 10.1007/s00208-012-0800-6
- Giovanni Catino, Carlo Mantegazza, and Lorenzo Mazzieri, Locally conformally flat ancient Ricci flows, Anal. PDE 8 (2015), no. 2, 365–371. MR 3345631, DOI 10.2140/apde.2015.8.365
- Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
- S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
- Manuel Fernández-López and Eduardo García-Río, Maximum principles and gradient Ricci solitons, J. Differential Equations 251 (2011), no. 1, 73–81. MR 2793264, DOI 10.1016/j.jde.2011.03.020
- Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
- Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
- Leon Karp, On Stokes’ theorem for noncompact manifolds, Proc. Amer. Math. Soc. 82 (1981), no. 3, 487–490. MR 612746, DOI 10.1090/S0002-9939-1981-0612746-9
- Paolo Mastrolia, Marco Rigoli, and Michele Rimoldi, Some geometric analysis on generic Ricci solitons, Commun. Contemp. Math. 15 (2013), no. 3, 1250058, 25. MR 3063556, DOI 10.1142/S0219199712500587
- Aaron Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010), 125–153. MR 2673425, DOI 10.1515/CRELLE.2010.062
- Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI 10.2307/2373587
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 [math.DG], 2002.
- G. Perelman, Ricci flow with surgery on three manifolds. arXiv:math/0303109v1 [math.DG], 2003.
- Peter Petersen and William Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2009), no. 2, 329–345. MR 2507581, DOI 10.2140/pjm.2009.241.329
- Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing and finiteness results in geometric analysis, Progress in Mathematics, vol. 266, Birkhäuser Verlag, Basel, 2008. A generalization of the Bochner technique. MR 2401291, DOI 10.1007/978-3-7643-8642-9
- Patrizia Pucci, Marco Rigoli, and James Serrin, Qualitative properties for solutions of singular elliptic inequalities on complete manifolds, J. Differential Equations 234 (2007), no. 2, 507–543. MR 2300666, DOI 10.1016/j.jde.2006.11.013
- Shun-ichi Tachibana, A theorem on Riemannian manifolds of positive curvature operator, Proc. Japan Acad. 50 (1974), 301–302. MR 365415
- Yoshihiro Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275. MR 174022, DOI 10.1090/S0002-9947-1965-0174022-6
- K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
Additional Information
- G. Catino
- Affiliation: Department of Mathematics, Politecnico di Milano, 20133 Milan, Italy
- MR Author ID: 887335
- Email: giovanni.catino@polimi.it
- P. Mastrolia
- Affiliation: Department of Mathematics, Università degli Studi di Milano, 20133 Milan, Italy
- MR Author ID: 896284
- 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
- MR Author ID: 881531
- Email: dario.monticelli@polimi.it
- M. Rigoli
- Affiliation: Department of Mathematics, Università degli Studi di Milano, 20133 Milan, Italy
- MR Author ID: 148315
- Email: marco.rigoli@unimi.it
- 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) - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7533-7549
- MSC (2010): Primary 53C20, 53C25
- DOI: https://doi.org/10.1090/tran/6864
- MathSciNet review: 3546774