Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 15, 2020

Positive solutions to Schrödinger equations and geometric applications

  • Ovidiu Munteanu , Felix Schulze ORCID logo EMAIL logo and Jiaping Wang


A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

Funding source: Leverhulme Trust

Award Identifier / Grant number: VP2-2018-029

Award Identifier / Grant number: RPG-2016-174

Award Identifier / Grant number: DMS-1506220

Funding statement: The first author was partially supported by NSF grant DMS-1506220 and by a Leverhulme Trust Visiting Professorship VP2-2018-029. The second author was supported by a Leverhulme Trust Research Project Grant RPG-2016-174.


[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. 10.1515/9781400853076Search in Google Scholar

[2] W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. 10.2307/1970868Search 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.a3Search in Google Scholar

[4] H.-D. Cao, Y. Shen and S. Zhu, The structure of stable minimal hypersurfaces in n + 1 , Math. Res. Lett. 4 (1997), no. 5, 637–644. 10.4310/MRL.1997.v4.n5.a2Search 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/1287580963Search 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.2011.07.022Search in Google Scholar

[7] G. Carron, L 2 -cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), no. 4, 613–639. 10.1007/s002080050310Search 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/1246888488Search 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.3160280303Search 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. 10.2307/2952459Search 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/1214459897Search in Google Scholar

[12] R. J. Conlon, A. Deruelle and S. Sun, Classification results for expanding and shrinking gradient Kähler–Ricci solitons, preprint (2019), Search 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.a4Search 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.2008.03.021Search 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.a2Search 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/s00039-011-0137-4Search in Google Scholar

[17] E. Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Math. 1635, Springer, Berlin 1996. 10.1007/BFb0092907Search 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.3160270601Search 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/1214438998Search 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-2015-0050Search 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.a4Search in Google Scholar

[22] P. Li, Geometric analysis, Cambridge Stud. Adv. Math. 134, Cambridge University Press, Cambridge 2012. 10.1017/CBO9781139105798Search 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/1214448079Search 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/1090348357Search 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.2006.11.001Search 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-ySearch in Google Scholar

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

[28] O. Munteanu and J. Wang, Geometry of shrinking Ricci solitons, Compos. Math. 151 (2015), no. 12, 2273–2300. 10.1112/S0010437X15007496Search 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/1427202765Search 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.2400Search in Google Scholar

[31] A. Naber, Noncompact shrinking four solitons with nonnegative curvature, J. reine angew. Math. 645 (2010), 125–153. 10.1515/crelle.2010.062Search 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. 10.1215/00127094-2795108Search in Google Scholar

[33] L. Ni, Gap theorems for minimal submanifolds in n + 1 , Comm. Anal. Geom. 9 (2001), no. 3, 641–656. 10.4310/CAG.2001.v9.n3.a2Search 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.a9Search in Google Scholar

[35] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002), Search 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.2005.05.007Search 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/1214448748Search 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/CBO9780511549762Search 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/BF01393992Search 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.a4Search in Google Scholar

[41] A. Sun and Z. Wang, Compactness of self-shrinkers in 3 with fixed genus, Adv. Math. 367 (2020), Article ID 107110. 10.1016/j.aim.2020.107110Search 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.a9Search 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/135Search 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-XSearch in Google Scholar

Received: 2020-07-23
Revised: 2020-10-12
Published Online: 2020-12-15
Published in Print: 2021-05-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.9.2023 from
Scroll to top button