Articles

ON THE DIMENSIONS OF SPACES OF HARMONIC FUNCTIONS WITH POLYNOMIAL GROWTH

  • Xiantao HUANG
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  • School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received date: 2018-07-27

  Revised date: 2019-01-24

  Online published: 2019-11-11

Supported by

The author is partially supported by NSFC (11701580 and 11521101) and the Fundamental Research Funds for the Central Universities (17lgpy13).

Abstract

In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.

Cite this article

Xiantao HUANG . ON THE DIMENSIONS OF SPACES OF HARMONIC FUNCTIONS WITH POLYNOMIAL GROWTH[J]. Acta mathematica scientia, Series B, 2019 , 39(5) : 1219 -1234 . DOI: 10.1007/s10473-019-0502-1

References

[1] Yau S-T. Harmonic functions on complete Riemannian manifolds. Comm Pure Appl Math, 1975, 28:201-228
[2] Cheng S Y. Liouville theorem for harmonic maps//Geometry of the Laplace operator:Proc Sympos Pure Math, XXXVI. Providence, R I:Amer Math Soc 1980:147-151
[3] Li P, Tam L-F. Linear growth harmonic functions on a complete manifold. J Differential Geom, 1989, 29(2):421-425
[4] Li P, Tam L-F. Complete surfaces with finite total curvature. J Differential Geom, 1991, 33(1):139-168
[5] Colding T H, Minicozzi W P II. Harmonic functions on manifolds. Ann Math, 1997, 146(3):725-747
[6] Colding T H, Minicozzi W P II. Liouville theorems for harmonic sections and applications. Comm Pure Appl Math, 1998, 51(2):113-138
[7] Colding T H, Minicozzi W P II. Weyl type bounds for harmonic functions. Invent Math, 1998, 131(2):257-298
[8] Li P. Harmonic sections of polynomial growth. Math Res Lett, 1997, 4(1):35-44
[9] Li P, Wang J. Counting massive sets and dimensions of harmonic functions. J Differential Geom, 1999, 53(2):237-278
[10] Ni L. The large time asymptotics of the entropy//Complex Analysis. Trends Math. Basel:Birkhäuser/Springer, 2010:301-306
[11] Ding Y. An existence theorem of harmonic functions with polynomial growth. Proc Amer Math Soc, 2004, 132(2):543-551
[12] Xu G. Three circles theorems for harmonic functions. Math Ann, 2016, 366(3/4):1281-1317
[13] Sormani C. Harmonic functions on manifolds with nonnegative Ricci curvature and linear volume growth. Pacific J Math, 2000, 192(1):183-189
[14] Huang X-T. On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth. J Reine Angew Math, 2018. https://doi.org/10.1515/crelle-2018-0029
[15] Honda S. Harmonic functions on asymptotic cones with Euclidean volume growth. J Math Soc Japan, 2015, 67(1):69-126
[16] Colding T H, Minicozzi W P II. Harmonic functions with polynomial growth. J Differential Geom, 1997, 46(1):1-77
[17] Ding Y. Heat kernels and Green's functions on limit spaces. Comm Anal Geom, 2002, 10(3):475-514
[18] Honda S. Ricci curvature and convergence of Lipschitz functions. Comm Anal Geom, 2011, 19(1):79-158
[19] Xu G. Large time behavior of the heat kernel. J Differential Geom, 2014, 98(3):467-528
[20] Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. I. J Differential Geom, 1997, 46(3):406-480
[21] Perelman G. A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone//Comparison Geometry:Math Sci Res Inst Publ, 30. Cambridge:Cambridge Univ Press, 1997:165-166
[22] Colding T H, Naber A. Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications. Geom Funct Anal, 2013, 23(1):134-148
[23] Cheeger J, Colding T H. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann Math, 1996, 144(1):189-237
[24] Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom Funct Anal, 1999, 9(3):428-517
[25] Shanmugalingam N. Newtonian spaces:an extension of Sobolev spaces to metric measure spaces. Rev Mat Iberoam, 2000, 16(2):243-279
[26] Ambrosio L, Gigli N, Savaré G. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent Math, 2014, 195(2):289-391
[27] Gigli N. On the differential structure of metric measure spaces and applications. Mem Amer Math Soc, 2015, 236(1113), vi+91 pp
[28] Li P, Schoen R. Lp and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math, 1984, 153(3/4):279-301
[29] Cheng S Y, Yau S-T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 28(3):333-354
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