, Volume 16, Issue 2, pp 453-475
Date: 04 May 2006

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

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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 M n with a trivial first homology group. The first upper bound will be in terms of the diameter of M n , 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 M n . 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 M n 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.

Received: October 2004 Revision: May 2005 Accepted: June 2005