Volume 13, issue 2 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Flats and the flat torus theorem in systolic spaces

Tomasz Elsner

Geometry & Topology 13 (2009) 661–698
Abstract

We prove the Systolic Flat Torus Theorem, which completes the list of basic properties that are simultaneously true for systolic geometry and CAT(0) geometry.

We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries.

Keywords
systolic complex, systolic group, minimal surface, flat, flat torus
Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 53C21
References
Publication
Received: 17 June 2007
Revised: 30 September 2008
Accepted: 25 October 2008
Preview posted: 11 December 2008
Published: 1 January 2009
Proposed: Martin Bridson
Seconded: Walter Neumann, Tobias Colding
Authors
Tomasz Elsner
Department of Mathematics
The Ohio State University
231 W 18th Ave
Columbus, OH 43210
USA
and Instytut Matematyczny
Uniwersytet Wrocławski
pl. Grunwaldzki 2/4
50-384 Wrocław
Poland
http://www.math.uni.wroc.pl/~elsner