Metric minimizing surfaces

Author:
Anton Petrunin

Journal:
Electron. Res. Announc. Amer. Math. Soc. **5** (1999), 47-54

MSC (1991):
Primary 53C21

Published electronically:
April 8, 1999

MathSciNet review:
1679453

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Abstract | References | Similar Articles | Additional Information

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****[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****[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**, 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**, 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****[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****[She1]**S. Z. Šefel′,*On saddle surfaces bounded by a rectifiable curve*, Dokl. Akad. Nauk SSSR**162**(1965), 294–296 (Russian). MR**0179700****[She2]**S. Z. Šefel′,*On the intrinsic geometry of saddle surfaces*, Sibirsk. Mat. Ž.**5**(1964), 1382–1396 (Russian). MR**0175046**

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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

Received by editor(s):
September 14, 1998

Published electronically:
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