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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main result can be stated roughly as follows: Let be an Alexandrov space, an open domain and a harmonic function. Then is Lipschitz on any compact subset of .

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

**[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**

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (2000):
51K10,
31B99

Retrieve articles in all journals with MSC (2000): 51K10, 31B99

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