A note on divergence of 𝐿^{𝑝}-integrals of subharmonic functions and its applications

Takegoshi, Kensho

doi:10.1090/S0002-9939-03-07042-4

A note on divergence of $L^{p}$-integrals of subharmonic functions and its applications
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by Kensho Takegoshi

Proc. Amer. Math. Soc. 131 (2003), 2849-2858

DOI: https://doi.org/10.1090/S0002-9939-03-07042-4

Published electronically: April 1, 2003

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Abstract:

A non $L^{p}$-integrability condition of non-constant non-negative subharmonic functions on a general complete manifold $(M,g)$ is given in an optimal form. As an application in differential geometry, several topics related to parabolicity of manifolds, the Liouville theorem for harmonic maps and conformal deformation of metrics are shown without any assumption on the Ricci curvature of $(M,g)$.

References

L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe type equations on complete manifolds and applications, J. Funct. Anal. 160 (1998), no. 1, 176–222. MR 1658696, DOI 10.1006/jfan.1998.3313
Kuo-Shung Cheng and Wei-Ming Ni, On the structure of the conformal scalar curvature equation on $\textbf {R}^n$, Indiana Univ. Math. J. 41 (1992), no. 1, 261–278. MR 1160913, DOI 10.1512/iumj.1992.41.41015
Hyeong In Choi and Andrejs Treibergs, Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space, J. Differential Geom. 32 (1990), no. 3, 775–817. MR 1078162
Shiu Yuen Cheng, Luen-Fai Tam, and Tom Y.-H. Wan, Harmonic maps with finite total energy, Proc. Amer. Math. Soc. 124 (1996), no. 1, 275–284. MR 1307503, DOI 10.1090/S0002-9939-96-03170-X
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. MR 385749, DOI 10.1002/cpa.3160280303
James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
A. A. Grigor′yan, On the fundamental solution of the heat equation on an arbitrary Riemannian manifold, Mat. Zametki 41 (1987), no. 5, 687–692, 765 (Russian). MR 898129
M. Gromov, Kähler hyperbolicity and $L_2$-Hodge theory, J. Differential Geom. 33 (1991), no. 1, 263–292. MR 1085144
R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983
R. E. Greene and H. Wu, $C^{\infty }$ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376
Leon Karp, Subharmonic functions on real and complex manifolds, Math. Z. 179 (1982), no. 4, 535–554. MR 652859, DOI 10.1007/BF01215065
Peter Li and Luen-Fai Tam, Complete surfaces with finite total curvature, J. Differential Geom. 33 (1991), no. 1, 139–168. MR 1085138
Peter Li and Luen-Fai Tam, Green’s functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), no. 2, 277–318. MR 1331970
Peter Li and Shing-Tung Yau, Curvature and holomorphic mappings of complete Kähler manifolds, Compositio Math. 73 (1990), no. 2, 125–144. MR 1046734
Karl-Theodor Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173–196. MR 1301456, DOI 10.1515/crll.1994.456.173
Kensh\B{o} Takegoshi, Energy estimates and Liouville theorems for harmonic maps, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 4, 563–592. MR 1072818
Kensho Takegoshi, A maximum principle for $P$-harmonic maps with $L^q$ finite energy, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3749–3753. MR 1469437, DOI 10.1090/S0002-9939-98-04609-7
N. T. Varopoulos, Potential theory and diffusion on Riemannian manifolds, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 821–837. MR 730112
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051