Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 5, 2020

Geometric and spectral estimates based on spectral Ricci curvature assumptions

Gilles Carron and Christian Rose

Abstract

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian ( M n , g ) manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator Δ + ( n - 2 ) ρ is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: CCEM-17-CE40-0034

Award Identifier / Grant number: ANR-11-LABX-0020-01

Funding statement: Gilles Carron was partially supported by the ANR grant CCEM-17-CE40-0034 and he wants to thank the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.

Acknowledgements

Christian Rose wants to thank the regional project DéfiMaths and Gilles Carron for their hospitality during his stay in Nantes, where parts of this work had been done. We also want to thank the referee for carefully reading the manuscript and for providing suggestions that improved the presentation.

References

[1] M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273. 10.1002/cpa.3160350206Search in Google Scholar

[2] E. Aubry, Finiteness of π 1 and geometric inequalities in almost positive Ricci curvature, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 4, 675–695. 10.1016/j.ansens.2007.07.001Search in Google Scholar

[3] E. Aubry, Bounds on the volume entropy and simplicial volume in Ricci curvature L p -bounded from below, Int. Math. Res. Not. IMRN 2009 (2009), no. 10, 1933–1946. 10.1093/imrn/rnp006Search in Google Scholar

[4] D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour 1992), Lecture Notes in Math. 1581, Springer, Berlin (1994), 1–114. 10.1007/BFb0073872Search in Google Scholar

[5] D. Bakry and M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), no. 1, 253–270. 10.1215/S0012-7094-96-08511-7Search in Google Scholar

[6] J. Barta, Sur la vibration fundamentale d’une membrane, C. R. Acad. Sci. 204 (1937), 472–473. Search in Google Scholar

[7] V. Bour and G. Carron, A sphere theorem for three dimensional manifolds with integral pinched curvature, Comm. Anal. Geom. 25 (2017), no. 1, 97–124. 10.4310/CAG.2017.v25.n1.a3Search in Google Scholar

[8] P. Buser, A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Supér. (4) 15 (1982), no. 2, 213–230. 10.24033/asens.1426Search in Google Scholar

[9] G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), no. 10, 883–891. 10.1016/S0021-7824(97)89976-XSearch in Google Scholar

[10] G. Carron, Geometric inequalities for manifolds with Ricci curvature in the Kato class, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 7, 3095–3167. 10.5802/aif.3346Search in Google Scholar

[11] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner 1969), Princeton University Press, Princeton (1970), 195–199. 10.1515/9781400869312-013Search in Google Scholar

[12] T. H. Colding and W. P. Minicozzi, II, Ricci curvature and monotonicity for harmonic functions, Calc. Var. Partial Differential Equations 49 (2014), no. 3–4, 1045–1059. 10.1007/s00526-013-0610-zSearch in Google Scholar

[13] X. Dai, P. Petersen and G. Wei, Integral pinching theorems, Manuscripta Math. 101 (2000), no. 2, 143–152. 10.1007/s002290050009Search in Google Scholar

[14] K. D. Elworthy and S. Rosenberg, Manifolds with wells of negative curvature. With an appendix by Daniel Ruberman, Invent. Math. 103 (1991), no. 3, 471–495. 10.1007/BF01239523Search in Google Scholar

[15] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. 10.1002/cpa.3160330206Search in Google Scholar

[16] S. Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, On the geometry of differentiable manifolds (Rome 1986), Astérisque 163–164, Société Mathématique de France, Paris (1988), 5–6 31–91, 281. Search in Google Scholar

[17] S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Colloque Paul Lévy sur les processus stochastiques (Palaiseau 1987), Astérisque 157–158, Société Mathématique de France, Paris (1988), 191–216. Search in Google Scholar

[18] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin 2015. Search in Google Scholar

[19] B. Güneysu, Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1289–1300. 10.1090/S0002-9939-2014-11859-4Search in Google Scholar

[20] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148. 10.1007/BF02760233Search in Google Scholar

[21] M. Ledoux, A simple analytic proof of an inequality by P. Buser, Proc. Amer. Math. Soc. 121 (1994), no. 3, 951–959. 10.1090/S0002-9939-1994-1186991-XSearch in Google Scholar

[22] I. Mondello, The local Yamabe constant of Einstein stratified spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 1, 249–275. 10.1016/j.anihpc.2015.12.001Search in Google Scholar

[23] W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), no. 1, 219–226. 10.2140/pjm.1978.75.219Search in Google Scholar

[24] P. Petersen, On eigenvalue pinching in positive Ricci curvature, Invent. Math. 138 (1999), no. 1, 1–21. 10.1007/s002220050339Search in Google Scholar

[25] P. Petersen, S. D. Shteingold and G. Wei, Comparison geometry with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1011–1030. 10.1007/s000390050035Search in Google Scholar

[26] P. Petersen and C. Sprouse, Integral curvature bounds, distance estimates and applications, J. Differential Geom. 50 (1998), no. 2, 269–298. 10.4310/jdg/1214461171Search in Google Scholar

[27] P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031–1045. 10.1007/s000390050036Search in Google Scholar

[28] P. Petersen and G. Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds. II, Trans. Amer. Math. Soc. 353 (2001), no. 2, 457–478. 10.1090/S0002-9947-00-02621-0Search in Google Scholar

[29] S. Pigola, M. Rigoli and A. G. Setti, Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique, Progr. Math. 266, Birkhäuser, Basel 2008. Search in Google Scholar

[30] C. Rose, Heat kernel estimates based on Ricci curvature integral bounds, Dissertation, Technische Universität Chemnitz, 2017. Search in Google Scholar

[31] C. Rose, Li–Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class, Ann. Global Anal. Geom. 55 (2019), no. 3, 443–449. 10.1007/s10455-018-9634-0Search in Google Scholar

[32] C. Rose and P. Stollmann, The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2199–2210. 10.1090/proc/13399Search in Google Scholar

[33] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. 10.1090/S0273-0979-1982-15041-8Search in Google Scholar

[34] C. Sprouse, Integral curvature bounds and bounded diameter, Comm. Anal. Geom. 8 (2000), no. 3, 531–543. 10.4310/CAG.2000.v8.n3.a4Search in Google Scholar

[35] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138. 10.1007/BF00396775Search in Google Scholar

[36] J. Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), no. 2, 167–205. 10.1016/0022-1236(86)90036-4Search in Google Scholar

[37] J.-G. Yun, A sphere theorem with integral curvature bounds, Kyushu J. Math. 56 (2002), no. 2, 225–234. 10.2206/kyushujm.56.225Search in Google Scholar

Received: 2018-10-10
Revised: 2020-05-30
Published Online: 2020-08-05
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston