Harmonic functions on Alexandrov spaces and their applications
Author:
Anton Petrunin
Translated by:
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 135141
MSC (2000):
Primary 51K10; Secondary 31B99
Published electronically:
December 17, 2003
MathSciNet review:
2030174
Fulltext PDF Free Access
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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]
F.
J. Almgren Jr., Existence and regularity almost everywhere of
solutions to elliptic variational problems with constraints, Mem.
Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 0420406
(54 #8420)
 [BGP]
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
(93m:53035), http://dx.doi.org/10.1070/RM1992v047n02ABEH000877
 [F]
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 [G]
Vitali
D. Milman and Gideon
Schechtman, Asymptotic theory of finitedimensional normed
spaces, Lecture Notes in Mathematics, vol. 1200, SpringerVerlag,
Berlin, 1986. With an appendix by M. Gromov. MR 856576
(87m:46038)
 [LU]
O.
A. Ladyzhenskaya and N.
N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo
tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second
edition, revised. MR 0509265
(58 #23009)
 [KMS1]
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
(2002g:53066), http://dx.doi.org/10.1090/conm/258/1778111
 [KMS2]
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
(2002m:58052), http://dx.doi.org/10.1007/s002090100252
 [N]
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
(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]
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
(94h:53055)
 [Pet1]
Petrunin, A., Subharmonic functions on Alexandrov space, preprint available at http://www.math.psu.edu/petrunin/.
 [Pet2]
A.
Petrunin, Parallel transportation for Alexandrov space with
curvature bounded below, Geom. Funct. Anal. 8 (1998),
no. 1, 123–148. MR 1601854
(98j:53048), http://dx.doi.org/10.1007/s000390050050
 [R]
Yu.
G. Reshetnyak, Twodimensional 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
(92b:53104)
 [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, 158 (1992); translation from Usp. Mat. Nauk 47, No. 2 (284), 351 (1992). MR 93m:53035
 [F]
 Federer, Herbert, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, SpringerVerlag 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), 275284, 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, 269316. 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, 114132. 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, 242256, Translation: St. Petersburg Math. J. 5 (1994), no. 1, 215227. 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, 123148 (1998). MR 98j:53048
 [R]
 Reshetnyak Yu. G., Twodimensional manifolds of bounded curvature. (English. Russian original) [CA] Geometry IV. Nonregular Riemannian geometry. Encycl. Math. Sci. 70, 3163 (1993); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 70, 7189 (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:
http://dx.doi.org/10.1090/S1079676203001203
PII:
S 10796762(03)001203
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 DMS0103957.
Communicated by:
Dmitri Burago
Article copyright:
© Copyright 2003
American Mathematical Society
