On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Xian-Tao Huang

doi:10.1515/crelle-2018-0029

Article

On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Xian-Tao Huang

Published/Copyright: November 14, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 762

Abstract

Suppose (Mn,g) is a Riemannian manifold with nonnegative Ricci curvature, and let hd⁢(M) be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that hd⁢(M) is finite. Later on, there are many researches which give better estimates of hd⁢(M). In this paper, we study the behavior of hd⁢(M) when d is large. More precisely, suppose (Mn,g) has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of hd⁢(M) in terms of the growth order d, the dimension n and the asymptotic volume ratio α=limR→∞⁡Vol⁢(Bp⁢(R))Rn. When α=ωn, i.e., (Mn,g) is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of hd⁢(ℝn).

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11701580

Award Identifier / Grant number: 11521101

Funding statement: The author is partially supported by NSFC 11701580 and NSFC 11521101.

Acknowledgements

The author would like to thank Professor B.-L. Chen, Professor H.-C. Zhang and Professor X.-P. Zhu and Dr. Y. Jiang for helpful discussions. The author would like to thank Professor Cheeger for comments. The author is grateful to the anonymous referees for careful reading and giving valuable suggestions.

References

[1] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4661–4701. 10.1090/S0002-9947-2015-06111-XSearch in Google Scholar

[2] L. Ambrosio, N. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), no. 2, 289–391. 10.1007/s00222-013-0456-1Search in Google Scholar

[3] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), no. 7, 1405–1490. 10.1215/00127094-2681605Search in Google Scholar

[4] L. Ambrosio and S. Honda, Local spectral convergence in RCD*⁢(K,N) spaces, Nonlinear Anal. 177 (2018), no. Part A, 1–23. 10.1016/j.na.2017.04.003Search in Google Scholar

[5] L. Ambrosio, S. Honda and D. Tewodrose, Short-time behavior of the heat kernel and Weyl’s law on RCD*⁢(K,N) spaces, Ann. Global Anal. Geom. 53 (2018), no. 1, 97–119. 10.1007/s10455-017-9569-xSearch in Google Scholar

[6] L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint (2015), https://arxiv.org/abs/1509.07273; to appear in Mem. Amer. Math. Soc. 10.1090/memo/1270Search in Google Scholar

[7] K. Bacher and K.-T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010), no. 1, 28–56. 10.1016/j.jfa.2010.03.024Search in Google Scholar

[8] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. 10.1007/s000390050094Search in Google Scholar

[9] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237. 10.2307/2118589Search in Google Scholar

[10] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. 10.4310/jdg/1214459974Search in Google Scholar

[11] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), no. 1, 13–35. 10.4310/jdg/1214342145Search in Google Scholar

[12] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37–74. 10.4310/jdg/1214342146Search in Google Scholar

[13] J. Cheeger, T. H. Colding and W. P. Minicozzi, II, Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature, Geom. Funct. Anal. 5 (1995), no. 6, 948–954. 10.1007/BF01902216Search in Google Scholar

[14] S. Y. Cheng, Liouville theorem for harmonic maps, Geometry of the Laplace operator (Honolulu 1979), Proc. Sympos. Pure Math. 36, American Mathematical Society, Providence (1980), 147–151. 10.1090/pspum/036/573431Search in Google Scholar

[15] 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

[16] 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

[17] T. H. Colding and W. P. Minicozzi, II, Weyl type bounds for harmonic functions, Invent. Math. 131 (1998), no. 2, 257–298. 10.1007/s002220050204Search in Google Scholar

[18] T. H. Colding and W. P. Minicozzi, II, On uniqueness of tangent cones for Einstein manifolds, Invent. Math. 196 (2014), no. 3, 515–588. 10.1007/s00222-013-0474-zSearch in Google Scholar

[19] T. H. Colding and A. Naber, Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. (2) 176 (2012), no. 2, 1173–1229. 10.4007/annals.2012.176.2.10Search in Google Scholar

[20] T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013), no. 1, 134–148. 10.1007/s00039-012-0202-7Search in Google Scholar

[21] G. De Philippis, A. Marchese and F. Rindler, On a conjecture of Cheeger, Measure theory in non-smooth spaces, Partial Differ. Equ. Meas. Theory, De Gruyter, Warsaw (2017), 145–155. 10.1515/9783110550832-004Search in Google Scholar

[22] Y. Ding, Heat kernels and Green’s functions on limit spaces, Comm. Anal. Geom. 10 (2002), no. 3, 475–514. 10.4310/CAG.2002.v10.n3.a3Search in Google Scholar

[23] Y. Ding, An existence theorem of harmonic functions with polynomial growth, Proc. Amer. Math. Soc. 132 (2004), no. 2, 543–551. 10.1090/S0002-9939-03-07060-6Search in Google Scholar

[24] H. Donnelly, Harmonic functions on manifolds of nonnegative Ricci curvature, Int. Math. Res. Not. IMRN 2002 (2001), no. 8, 429–434. 10.1155/S1073792801000216Search in Google Scholar

[25] M. Erbar, K. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), no. 3, 993–1071. 10.1007/s00222-014-0563-7Search in Google Scholar

[26] L. C. Evans, Partial differential equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence 2010. Search in Google Scholar

[27] N. Gigli, The splitting theorem in non-smooth context, preprint (2013), https://arxiv.org/abs/1302.5555. Search in Google Scholar

[28] N. Gigli, On the differential structure of metric measure spaces and applications, Mem. Amer. Math. Soc. 236 (2015), no. 1113, 1–91. 10.1090/memo/1113Search in Google Scholar

[29] N. Gigli and B.-X. Han, Sobolev spaces on warped products, J. Funct. Anal. 275 (2018), no. 8, 2059–2095. 10.1016/j.jfa.2018.03.021Search in Google Scholar

[30] N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071–1129. 10.1112/plms/pdv047Search in Google Scholar

[31] N. Gigli and E. Pasqualetto, Behaviour of the reference measure on RCD spaces under charts, preprint (2016), https://arxiv.org/abs/1607.05188. 10.4310/CAG.2021.v29.n6.a3Search in Google Scholar

[32] S. Honda, Ricci curvature and convergence of Lipschitz functions, Comm. Anal. Geom. 19 (2011), no. 1, 79–158. 10.4310/CAG.2011.v19.n1.a4Search in Google Scholar

[33] S. Honda, Harmonic functions on asymptotic cones with Euclidean volume growth, J. Math. Soc. Japan 67 (2015), no. 1, 69–126. 10.2969/jmsj/06710069Search in Google Scholar

[34] B. Hua, M. Kell and C. Xia, Harmonic functions on metric measure spaces, preprint (2013), https://arxiv.org/abs/1308.3607. Search in Google Scholar

[35] X.-T. Huang, On the dimensions of spaces of harmonic functions with polynomial growth, preprint (2016); to appear in Acta Math. Sci. Ser. B. 10.1007/s10473-019-0502-1Search in Google Scholar

[36] R. Jiang, Cheeger-harmonic functions in metric measure spaces revisited, J. Funct. Anal. 266 (2014), no. 3, 1373–1394. 10.1016/j.jfa.2013.11.022Search in Google Scholar

[37] M. Kell and A. Mondino, On the volume measure of non-smooth spaces with Ricci curvature bounded below, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 593–610. 10.2422/2036-2145.201608_007Search in Google Scholar

[38] C. Ketterer, Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl. (9) 103 (2015), no. 5, 1228–1275. 10.1016/j.matpur.2014.10.011Search in Google Scholar

[39] 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

[40] P. Li, Geometric analysis, Cambridge Stud. Adv. Math. 134, Cambridge University Press, Cambridge 2012. 10.1017/CBO9781139105798Search in Google Scholar

[41] P. Li and R. Schoen, Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), no. 3–4, 279–301. 10.1007/BF02392380Search in Google Scholar

[42] P. Li and L.-F. Tam, Linear growth harmonic functions on a complete manifold, J. Differential Geom. 29 (1989), no. 2, 421–425. 10.4310/jdg/1214442883Search in Google Scholar

[43] P. Li and L.-F. Tam, Complete surfaces with finite total curvature, J. Differential Geom. 33 (1991), no. 1, 139–168. 10.4310/jdg/1214446033Search in Google Scholar

[44] P. Li and J. Wang, Counting massive sets and dimensions of harmonic functions, J. Differential Geom. 53 (1999), no. 2, 237–278. 10.4310/jdg/1214425536Search in Google Scholar

[45] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. 10.4007/annals.2009.169.903Search in Google Scholar

[46] A. Mondino and A. Naber, Structure theory of metric-measure spaces with lower Ricci curvature bounds, preprint (2014), https://arxiv.org/abs/1405.2222; to appear in J. Eur. Math. Soc. 10.4171/JEMS/874Search in Google Scholar

[47] S.-I. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 211–249. 10.1007/s00526-009-0227-4Search in Google Scholar

[48] G. Perelman, A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, Comparison geometry (Berkeley 1993–94), Math. Sci. Res. Inst. Publ. 30, Cambridge University Press, Cambridge (1997), 165–166. Search in Google Scholar

[49] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. 10.1007/s11511-006-0002-8Search in Google Scholar

[50] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. 10.1007/s11511-006-0003-7Search in Google Scholar

[51] G. Xu, Large time behavior of the heat kernel, J. Differential Geom. 98 (2014), no. 3, 467–528. 10.4310/jdg/1406552278Search in Google Scholar

[52] G. Xu, Three circles theorems for harmonic functions, Math. Ann. 366 (2016), no. 3–4, 1281–1317. 10.1007/s00208-016-1366-5Search in Google Scholar

[53] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. 10.1002/cpa.3160280203Search in Google Scholar

[54] H. C. Zhang and X. P. Zhu, Weyl’s law on RCD*⁢(K,N) metric measure space, preprint (2017), https://arxiv.org/abs/1701.01967; to appear in Comm. Anal. Geom. Search in Google Scholar

Received: 2018-10-23

Revised: 2018-10-24

Published Online: 2018-11-14

Published in Print: 2020-05-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2018-0029