Geometric & Functional Analysis GAFA

, Volume 16, Issue 2, pp 453–475

Curvature-free upper bounds for the smallest area of a minimal surface

Original Paper

DOI: 10.1007/s00039-006-0559-6

Cite this article as:
Nabutovsky, A. & Rotman, R. GAFA, Geom. funct. anal. (2006) 16: 453. doi:10.1007/s00039-006-0559-6

Abstract.

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mn with a trivial first homology group. The first upper bound will be in terms of the diameter of Mn, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mn. If n  =  3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mn with \(H_{1} {\left( {M^{n} } \right)} = \cdots = H_{{k - 1}} {\left( {M^{n} } \right)} = {\left\{ 0 \right\}},{\left( {k > 2} \right)} \). The above results are the first results of such nature.

No Keywords and phrases.

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No 2000 Mathematics Subject Classification.

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada