Analytic and geometric properties of generic Ricci solitons

Catino, G.; Mastrolia, P.; Monticelli, D.; Rigoli, M.

doi:10.1090/tran/6864

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