Regina Rotman and Alexander Nabutovsky
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 $M^n$. 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 $H_1(M^n)=...=H_{k-1}(M^n)= \{ 0 \}$, $(k>2)$. The above results are the first results of such nature.