Bounding volume of compact domains via mean curvature and bottom spectrum
-
and
Abstract
The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1811845
Funding statement: The first author was partially supported by the NSF grant DMS-1811845 and by a Simons Foundation grant.
Acknowledgements
We thank Pengfei Guan and Luca Benatti for their interest in this work and for many valuable comments.
References
[1] V. Agostiniani, M. Fogagnolo and L. Mazzieri, Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature, Invent. Math. 222 (2020), no. 3, 1033–1101. 10.1007/s00222-020-00985-4Search in Google Scholar
[2] V. Agostiniani, M. Fogagnolo and L. Mazzieri, Minkowski inequalities via nonlinear potential theory, Arch. Ration. Mech. Anal. 244 (2022), no. 1, 51–85. 10.1007/s00205-022-01756-6Search in Google Scholar
[3] L. Benatti, M. Fogagnolo and L. Mazzieri, Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature, Anal. PDE 17 (2024), no. 9, 3039–3077. 10.2140/apde.2024.17.3039Search in Google Scholar
[4] S. Brendle, P.-K. Hung and M.-T. Wang, A Minkowski inequality for hypersurfaces in the anti-de Sitter–Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124–144. 10.1002/cpa.21556Search in Google Scholar
[5] R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981), no. 4, 581–598. 10.1007/BF02566228Search in Google Scholar
[6] G. Carron and E. Pedon, On the differential form spectrum of hyperbolic manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 705–747. 10.2422/2036-2145.2004.4.03Search in Google Scholar
[7] S.-Y. A. Chang and Y. Wang, Inequalities for quermassintegrals on 𝑘-convex domains, Adv. Math. 248 (2013), 335–377. 10.1016/j.aim.2013.08.006Search in Google Scholar
[8] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971/72), 119–128. 10.4310/jdg/1214430220Search in Google Scholar
[9] 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
[10] S. Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297. 10.1007/BF01214381Search in Google Scholar
[11] O. Chodosh, M. Eichmair and T. Koerber, On the Minkowski inequality near the sphere, preprint (2023), https://arxiv.org/abs/2306.03848. Search in Google Scholar
[12] B. Chow, Ricci solitons in low dimensions, Grad. Stud. Math. 235, American Mathematical Society, Providence 2023. 10.1090/gsm/235Search in Google Scholar
[13] B. Chow, B. Kotschwar and O. Munteanu, Ricci solitons in dimensions 4 and higher, Math. Surveys Monogr. 293, American Mathematical Society, Providence 2025. 10.1090/surv/293Search in Google Scholar
[14] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Flow, Grad. Stud. Math. 77, American Mathematical Society, Providence 2006. Search in Google Scholar
[15] T. H. Colding, New monotonicity formulas for Ricci curvature and applications. I, Acta Math. 209 (2012), no. 2, 229–263. 10.1007/s11511-012-0086-2Search in Google Scholar
[16] T. H. Colding and W. Minicozzi, Ricci curvature and monotonicity for harmonic functions, Calc. Var. Partial Differential Equations 49 (2014), 1045–1059. 10.1007/s00526-013-0610-zSearch in Google Scholar
[17] J. Dalphin, A. Henrot, S. Masnou and T. Takahashi, On the minimization of total mean curvature, J. Geom. Anal. 26 (2016), no. 4, 2729–2750. 10.1007/s12220-015-9646-ySearch in Google Scholar
[18] L. L. de Lima and F. Girão, An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality, Ann. Henri Poincaré 17 (2016), no. 4, 979–1002. 10.1007/s00023-015-0414-0Search in Google Scholar
[19]
J. Dodziuk,
[20] 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
[21] M. P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140–145. 10.2307/1969703Search in Google Scholar
[22] Y. Ge, G. Wang and J. Wu, Hyperbolic Alexandrov–Fenchel quermassintegral inequalities II, J. Differential Geom. 98 (2014), no. 2, 237–260. 10.4310/jdg/1406552250Search in Google Scholar
[23] M. Ghomi and J. Spruck, Total mean curvatures of Riemannian hypersurfaces, Adv. Nonlinear Stud. 23 (2023), no. 1, Article ID 20220029. 10.1515/ans-2022-0029Search in Google Scholar
[24] F. Glaudo, Minkowski inequality for nearly spherical domains, Adv. Math. 408 (2022), Article ID 108595. 10.1016/j.aim.2022.108595Search in Google Scholar
[25] P. Guan and J. Li, The quermassintegral inequalities for 𝑘-convex starshaped domains, Adv. Math. 221 (2009), no. 5, 1725–1732. 10.1016/j.aim.2009.03.005Search in Google Scholar
[26] P. Guan and J. Li, A mean curvature type flow in space forms, Int. Math. Res. Not. IMRN 2015 (2015), no. 13, 4716–4740. 10.1093/imrn/rnu081Search in Google Scholar
[27] P. Guan, J. Li and M.-T. Wang, A volume preserving flow and the isoperimetric problem in warped product spaces, Trans. Amer. Math. Soc. 372 (2019), no. 4, 2777–2798. 10.1090/tran/7661Search in Google Scholar
[28] R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry Vol. II (Cambridge 1993), International Press, Cambridge (1995), 7–136. 10.4310/SDG.1993.v2.n1.a2Search in Google Scholar
[29] K. Kunikawa and S. Saito, Remarks on topology of stable translating solitons, Geom. Dedicata 202 (2019), 1–8. 10.1007/s10711-018-0399-1Search in Google Scholar
[30] J. M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom. 3 (1995), no. 1–2, 253–271. 10.4310/CAG.1995.v3.n2.a2Search in Google Scholar
[31] P. Li, Geometric analysis, Cambridge Stud. Adv. Math. 134, Cambridge University, Cambridge 2012. 10.7202/1018492arSearch in Google Scholar
[32] 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
[33] P. Li and J. Wang, Complete manifolds with positive spectrum. II, J. Differential Geom. 62 (2002), no. 1, 143–162. 10.4310/jdg/1090425532Search in Google Scholar
[34] 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
[35] S. McCormick, On a Minkowski-like inequality for asymptotically flat static manifolds, Proc. Amer. Math. Soc. 146 (2018), no. 9, 4039–4046. 10.1090/proc/14047Search in Google Scholar
[36] H. P. McKean, An upper bound to the spectrum of Δ on a manifold of negative curvature, J. Differential Geom. 4 (1970), 359–366. 10.4310/jdg/1214429509Search in Google Scholar
[37] H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), no. 4, 447–495. 10.1007/BF01445180Search in Google Scholar
[38] O. Munteanu and J. Wang, Smooth metric measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011), no. 3, 451–486. 10.4310/CAG.2011.v19.n3.a1Search in Google Scholar
[39] O. Munteanu and J. Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55–94. 10.4310/CAG.2012.v20.n1.a3Search in Google Scholar
[40] O. Munteanu and J. Wang, Comparison theorems for 3D manifolds with scalar curvature bound, Int. Math. Res. Not. IMRN 2023 (2023), no. 3, 2215–2242. 10.1093/imrn/rnab307Search in Google Scholar
[41] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3–4, 241–273. 10.1007/BF02392046Search in Google Scholar
[42] S. Pigola, M. Rimoldi and A. G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011), no. 3–4, 777–790. 10.1007/s00209-010-0695-4Search in Google Scholar
[43] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327–351. 10.4310/jdg/1214440979Search in Google Scholar
[44] X. Wang, A new proof of Lee’s theorem on the spectrum of conformally compact Einstein manifolds, Comm. Anal. Geom. 10 (2002), no. 3, 647–651. 10.4310/CAG.2002.v10.n3.a7Search in Google Scholar
[45] Y. Wei, On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 46. 10.1007/s00526-018-1342-xSearch in Google Scholar
[46] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. 10.1002/cpa.3160280203Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Equations involving the modular 𝑗-function and its derivatives
- Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle
- Rigidity and compactness with constant mean curvature in warped product manifolds
- Classification of torsion of elliptic curves over quartic fields
- A new monotone quantity in mean curvature flow implying sharp homotopic criteria
- Volumes of subvarieties of complex ball quotients and sparsity of rational points
- Bounding volume of compact domains via mean curvature and bottom spectrum
- Some obstructions to positive scalar curvature on a noncompact manifold
Articles in the same Issue
- Frontmatter
- Equations involving the modular 𝑗-function and its derivatives
- Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle
- Rigidity and compactness with constant mean curvature in warped product manifolds
- Classification of torsion of elliptic curves over quartic fields
- A new monotone quantity in mean curvature flow implying sharp homotopic criteria
- Volumes of subvarieties of complex ball quotients and sparsity of rational points
- Bounding volume of compact domains via mean curvature and bottom spectrum
- Some obstructions to positive scalar curvature on a noncompact manifold
