Harmonic functions on Alexandrov spaces and their applications
Author:
Anton Petrunin
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
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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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- Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
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- Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Beginning of analysis on Alexandrov spaces, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 275–284. MR 1778111, DOI 10.1090/conm/258/1778111
- Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269–316. MR 1865418, DOI 10.1007/s002090100252
- I. G. Nikolaev, Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov, Sibirsk. Mat. Zh. 24 (1983), no. 2, 114–132 (Russian). MR 695295
[Per1]Per1 Perelman, G., DC structure on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
[Per2]Per2 Perelman, G., A. D. Alexandrov space with curvature bounded from below II, preprint.
- G. Ya. Perel′man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993), no. 1, 242–256 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 215–227. MR 1220499
[Pet1]Pet1 Petrunin, A., Subharmonic functions on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
- A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), no. 1, 123–148. MR 1601854, DOI 10.1007/s000390050050
- Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 7–189, 273–277, 279 (Russian). MR 1099202
[A]A Almgren, F., Existence and regularity almost everywhere of solutions to elliptic variational problem with constraints, Mem. A.M.S., 4 (1976).
[BGP]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).
[F]F Federer, Herbert, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, xiv+676 pp.
[G]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.
[LU]LU Ladyzhenskaya, O. A., Uraltseva, N. N., Linear and quasilinear equations of elliptic type, “Nauka”, Moscow, 1973. 576 pp.
[KMS1]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.
[KMS2]KMS2 Kuwae, Kazuhiro, Machigashira, Yoshiroh, Shioya, Takashi, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269–316.
[N]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.
[Per1]Per1 Perelman, G., DC structure on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
[Per2]Per2 Perelman, G., A. D. Alexandrov space with curvature bounded from below II, preprint.
[PerPet]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.
[Pet1]Pet1 Petrunin, A., Subharmonic functions on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
[Pet2]Pet2 Petrunin, A., Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8, No. 1, 123-148 (1998).
[R]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).
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Additional Information
Anton Petrunin
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
MR Author ID:
335143
ORCID:
0000-0003-3053-5172
Email:
petrunin@math.psu.edu
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
