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Harmonic functions on Alexandrov spaces and their applications


Author: Anton Petrunin
Translated by:
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 135-141
MSC (2000): Primary 51K10; Secondary 31B99
DOI: https://doi.org/10.1090/S1079-6762-03-00120-3
Published electronically: December 17, 2003
MathSciNet review: 2030174
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Abstract | References | Similar Articles | Additional Information

Abstract: The main result can be stated roughly as follows: Let $M$ be an Alexandrov space, $\Omega \subset M$ an open domain and $f:\Omega \to \mathbb{R}$ a harmonic function. Then $f$ is Lipschitz on any compact subset of $\Omega $.

Using this result I extend proofs of some classical theorems in Riemannian geometry to Alexandrov spaces.


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  • [A] Almgren, F., Existence and regularity almost everywhere of solutions to elliptic variational problem with constraints, Mem. A.M.S., 4 (1976). MR 54:8420
  • [BGP] Burago, Yu., Gromov, M., Perelman, G., A. D. Alexandrov spaces with curvature bounded below. (English. Russian original), Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2 (284), 3-51 (1992). MR 93m:53035
  • [F] Federer, Herbert, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, xiv+676 pp. MR 41:1976
  • [G] Gromov, M., Isoperimetric inequalities in Riemannian manifolds, appendix in Milman, Vitali D., Schechtman, Gideon, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Mathematics, 1200. MR 87m:46038
  • [LU] Ladyzhenskaya, O. A., Uraltseva, N. N., Linear and quasilinear equations of elliptic type, ``Nauka'', Moscow, 1973. 576 pp. MR 58:23009
  • [KMS1] Kuwae, Kazuhiro, Machigashira, Yoshiroh, Shioya, Takashi, Beginning of analysis on Alexandrov spaces, Geometry and topology: Aarhus (1998), 275-284, Contemp. Math., 258, Amer. Math. Soc., Providence, RI, 2000. MR 2002g:53066
  • [KMS2] Kuwae, Kazuhiro, Machigashira, Yoshiroh, Shioya, Takashi, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316. MR 2002m:58052
  • [N] Nikolaev, I. G., Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov, (Russian) Sibirsk. Mat. Zh. 24 (1983), no. 2, 114-132. MR 84h:53098
  • [Per1] Perelman, G., DC structure on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
  • [Per2] Perelman, G., A. D. Alexandrov space with curvature bounded from below II, preprint.
  • [PerPet] Perelman G. Ya., Petrunin A. M., Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993), no. 1, 242-256, Translation: St. Petersburg Math. J. 5 (1994), no. 1, 215-227. MR 94h:53055
  • [Pet1] Petrunin, A., Subharmonic functions on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
  • [Pet2] Petrunin, A., Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8, No. 1, 123-148 (1998). MR 98j:53048
  • [R] Reshetnyak Yu. G., Two-dimensional manifolds of bounded curvature. (English. Russian original) [CA] Geometry IV. Nonregular Riemannian geometry. Encycl. Math. Sci. 70, 3-163 (1993); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 70, 7-189 (1989). MR 92b:53104

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Additional Information

Anton Petrunin
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email: petrunin@math.psu.edu

DOI: https://doi.org/10.1090/S1079-6762-03-00120-3
Received by editor(s): March 4, 2003
Published electronically: December 17, 2003
Additional Notes: The main part of this paper was written while I had postdoctoral fellowship at MSRI in 1995–1996. I would like to thank this institute for providing excellent conditions to conduct this research. I was also supported by NSF DMS-0103957.
Communicated by: Dmitri Burago
Article copyright: © Copyright 2003 American Mathematical Society

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