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

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.