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Metric minimizing surfaces
Author:
Anton Petrunin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 47-54
MSC (1991):
Primary 53C21
Posted:
April 8, 1999
MathSciNet review:
1679453
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Abstract: Consider a two-dimensional surface in an Alexandrov space of curvature bounded above by  . Assume that this surface does not admit contracting deformations (a particular case of such surfaces is formed by area minimizing surfaces). Then this surface inherits the upper curvature bound, that is, this surface has also curvature bounded above by  , with respect to the intrinsic metric induced from its ambient space.
- [A]
A.
D. Aleksandrov, Ruled surfaces in metric spaces, Vestnik
Leningrad. Univ. 12 (1957), no. 1, 5–26, 207
(Russian, with English summary). MR 0085564
(19,59a)
- [GLP]
Mikhael
Gromov, Structures métriques pour les variétés
riemanniennes, Textes Mathématiques [Mathematical Texts],
vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P.
Pansu. MR
682063 (85e:53051)
- [M]
C. Mese, The curvature of minimal surfaces in singular spaces, to appear in Comm. Anal. Geom.
- [N]
Igor
Nikolaev, The tangent cone of an Aleksandrov space of curvature
≤𝐾, Manuscripta Math. 86 (1995),
no. 2, 137–147. MR 1317739
(95m:53062), http://dx.doi.org/10.1007/BF02567983
- [P]
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]
Ju.
G. Rešetnjak, Non-expansive maps in a space of curvature no
greater than 𝐾, Sibirsk. Mat. Ž. 9
(1968), 918–927 (Russian). MR 0244922
(39 #6235)
- [Sha]
V.
A. Šarafutdinov, Radius of injectivity of a complete open
manifold of nonnegative curvature, Dokl. Akad. Nauk SSSR
231 (1976), no. 1, 46–48 (Russian). MR 0451172
(56 #9459)
- [She1]
S.
Z. Šefel′, On saddle surfaces bounded by a rectifiable
curve, Dokl. Akad. Nauk SSSR 162 (1965),
294–296 (Russian). MR 0179700
(31 #3945)
- [She2]
S.
Z. Šefel′, On the intrinsic geometry of saddle
surfaces, Sibirsk. Mat. Ž. 5 (1964),
1382–1396 (Russian). MR 0175046
(30 #5232)
- [A]
- A.D. Aleksandrov, Ruled surfaces in metric spaces, Vestnik LGU Ser. Mat., Mech., Astr. 1957, vyp. 1, 5-26. (Russian) MR 19:59a
- [GLP]
- M. Gromov, Structures métriques pour les variétés riemanniennes, J. Lafontaine and P. Pansu, eds., CEDIC, Paris, 1981. MR 85e:53051
- [M]
- C. Mese, The curvature of minimal surfaces in singular spaces, to appear in Comm. Anal. Geom.
- [N]
- I. Nikolaev, The tangent cone of an Aleksandrov space of curvature
 , Manuscr. Math. 86 (1995), No. 2, 137-147. MR 95m:53062
- [P]
- A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), No. 1, 123-148. MR 98j:53048
- [R]
- Yu.G. Reshetniak, Non-expansive maps in a space of curvature no greater than
 , Sibirskii Mat. Zh. 9 (1968), 918-928; English transl., Siberian Math. J. 9 (1968), 683-689. MR 39:6235
- [Sha]
- V.A. Sharafutdinov, The radius of injectivity of a complete open manifold of nonnegative curvature, Doklady Akad. Nauk SSSR 231 (1976), No. 1, 46-48. MR 56:9459
- [She1]
- S.Z. Shefel, On saddle surfaces bounded by a rectifiable curve, Doklady Akad. Nauk SSSR 162 (1965), 294-296; English transl. in Soviet Math. Dokl. 6 (1965). MR 31:3945
- [She2]
- S.Z. Shefel, On intrinsic geometry of saddle surfaces, Sibirsk. Mat. Zh. 5 (1964), 1382-1396. (Russian) MR 30:5232
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Additional Information
Anton Petrunin
Affiliation:
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22-26, D-04103 Leipzig, Germany
Email:
petrunin@mailhost.mis.mpg.de
DOI:
http://dx.doi.org/10.1090/S1079-6762-99-00059-1
PII:
S 1079-6762(99)00059-1
Received by editor(s):
September 14, 1998
Posted:
April 8, 1999
Additional Notes:
The main part of this note was prepared when the author had a postdoctoral fellowship at MSRI (Berkeley).
Communicated by:
Dmitri Burago
Article copyright:
© Copyright 1999 American Mathematical Society
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