Positive solutions to Schrödinger equations and geometric applications

References

[1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: Bounds on eigenfunctions of N-body Schrödinger operators, Math. Notes 29, Princeton University Press, Princeton 1982. 10.1515/9781400853076Suche in Google Scholar

[2] W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. 10.2307/1970868Suche in Google Scholar

[3] H.-D. Cao, B.-L. Chen and X.-P. Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII: Geometric flows, Surv. Differ. Geom. 12, International Press, Somerville (2008), 47–112. 10.4310/SDG.2007.v12.n1.a3Suche in Google Scholar

[4] H.-D. Cao, Y. Shen and S. Zhu, The structure of stable minimal hypersurfaces in ℝ n + 1 , Math. Res. Lett. 4 (1997), no. 5, 637–644. 10.4310/MRL.1997.v4.n5.a2Suche in Google Scholar

[5] H.-D. Cao and D. Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. 10.4310/jdg/1287580963Suche in Google Scholar

[6] X. Cao and Q. S. Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228 (2011), no. 5, 2891–2919. 10.1016/j.aim.2011.07.022Suche in Google Scholar

[7] G. Carron, L 2 -cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), no. 4, 613–639. 10.1007/s002080050310Suche in Google Scholar

[8] B.-L. Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. 10.4310/jdg/1246888488Suche in Google Scholar

[9] 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. 10.1002/cpa.3160280303Suche in Google Scholar

[10] T. H. Colding and W. P. Minicozzi II, Harmonic functions on manifolds, Ann. of Math. (2) 146 (1997), no. 3, 725–747. 10.2307/2952459Suche in Google Scholar

[11] T. H. Colding and W. P. Minicozzi II, Harmonic functions with polynomial growth, J. Differential Geom. 46 (1997), no. 1, 1–77. 10.4310/jdg/1214459897Suche in Google Scholar

[12] R. J. Conlon, A. Deruelle and S. Sun, Classification results for expanding and shrinking gradient Kähler–Ricci solitons, preprint (2019), https://arxiv.org/abs/1904.00147v2. Suche in Google Scholar

[13] J. Enders, R. Müller and P. M. Topping, On type-I singularities in Ricci flow, Comm. Anal. Geom. 19 (2011), no. 5, 905–922. 10.4310/CAG.2011.v19.n5.a4Suche in Google Scholar

[14] F. Fang, J. Man and Z. Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C. R. Math. Acad. Sci. Paris 346 (2008), no. 11–12, 653–656. 10.1016/j.crma.2008.03.021Suche in Google Scholar

[15] R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), International Press, Cambridge (1995), 7–136. 10.4310/SDG.1993.v2.n1.a2Suche in Google Scholar

[16] R. Haslhofer and R. Müller, A compactness theorem for complete Ricci shrinkers, Geom. Funct. Anal. 21 (2011), no. 5, 1091–1116. 10.1007/s00039-011-0137-4Suche in Google Scholar

[17] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Math. 1635, Springer, Berlin 1996. 10.1007/BFb0092907Suche in Google Scholar

[18] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715–727. 10.1002/cpa.3160270601Suche in Google Scholar

[19] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. 10.4310/jdg/1214438998Suche in Google Scholar

[20] N. Kapouleas, S. J. Kleene and N. M. Møller, Mean curvature self-shrinkers of high genus: Non-compact examples, J. reine angew. Math. 739 (2018), 1–39. 10.1515/crelle-2015-0050Suche in Google Scholar

[21] P. Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), no. 1, 35–44. 10.4310/MRL.1997.v4.n1.a4Suche in Google Scholar

[22] P. Li, Geometric analysis, Cambridge Stud. Adv. Math. 134, Cambridge University Press, Cambridge 2012. 10.1017/CBO9781139105798Suche in Google Scholar

[23] P. Li and L.-F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359–383. 10.4310/jdg/1214448079Suche in Google Scholar

[24] P. Li and J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501–534. 10.4310/jdg/1090348357Suche in Google Scholar

[25] P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), no. 6, 921–982. 10.1016/j.ansens.2006.11.001Suche in Google Scholar

[26] Y. Li and B. Wang, Heat kernel on Ricci shrinkers, Calc. Var. Partial Differential Equations 59 (2020), no. 6, 194–194. 10.1007/s00526-020-01861-ySuche in Google Scholar

[27] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of ℝ n , Comm. Pure Appl. Math. 26 (1973), 361–379. 10.1002/cpa.3160260305Suche in Google Scholar

[28] O. Munteanu and J. Wang, Geometry of shrinking Ricci solitons, Compos. Math. 151 (2015), no. 12, 2273–2300. 10.1112/S0010437X15007496Suche in Google Scholar

[29] O. Munteanu and J. Wang, Topology of Kähler Ricci solitons, J. Differential Geom. 100 (2015), no. 1, 109–128. 10.4310/jdg/1427202765Suche in Google Scholar

[30] O. Munteanu and J. Wang, Structure at infinity for shrinking Ricci solitons, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 4, 891–925. 10.24033/asens.2400Suche in Google Scholar

[31] A. Naber, Noncompact shrinking four solitons with nonnegative curvature, J. reine angew. Math. 645 (2010), 125–153. 10.1515/crelle.2010.062Suche in Google Scholar

[32] X. H. Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. Part III, Duke Math. J. 163 (2014), no. 11, 2023–2056. 10.1215/00127094-2795108Suche in Google Scholar

[33] L. Ni, Gap theorems for minimal submanifolds in ℝ n + 1 , Comm. Anal. Geom. 9 (2001), no. 3, 641–656. 10.4310/CAG.2001.v9.n3.a2Suche in Google Scholar

[34] L. Ni and N. Wallach, On a classification of gradient shrinking solitons, Math. Res. Lett. 15 (2008), no. 5, 941–955. 10.4310/MRL.2008.v15.n5.a9Suche in Google Scholar

[35] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002), https://arxiv.org/abs/math/0211159. Suche in Google Scholar

[36] S. Pigola, M. Rigoli and A. G. Setti, Vanishing theorems on Riemannian manifolds, and geometric applications, J. Funct. Anal. 229 (2005), no. 2, 424–461. 10.1016/j.jfa.2005.05.007Suche in Google Scholar

[37] L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), no. 2, 417–450. 10.4310/jdg/1214448748Suche in Google Scholar

[38] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Math. Soc. Lecture Note Ser. 289, Cambridge University Press, Cambridge 2002. 10.1017/CBO9780511549762Suche in Google Scholar

[39] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. 10.1007/BF01393992Suche in Google Scholar

[40] N. Sesum, Convergence of the Ricci flow toward a soliton, Comm. Anal. Geom. 14 (2006), no. 2, 283–343. 10.4310/CAG.2006.v14.n2.a4Suche in Google Scholar

[41] A. Sun and Z. Wang, Compactness of self-shrinkers in ℝ 3 with fixed genus, Adv. Math. 367 (2020), Article ID 107110. 10.1016/j.aim.2020.107110Suche in Google Scholar

[42] P. Topping, Diameter control under Ricci flow, Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055. 10.4310/CAG.2005.v13.n5.a9Suche in Google Scholar

[43] P. Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539–546. 10.4171/CMH/135Suche in Google Scholar

[44] L. Wang, Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014), no. 3, 613–638. 10.1090/S0894-0347-2014-00792-XSuche in Google Scholar