Metric minimizing surfaces

Author:
Anton Petrunin

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

MSC (1991):
Primary 53C21

DOI:
https://doi.org/10.1090/S1079-6762-99-00059-1

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 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:
https://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