Curvature-free upper bounds for the smallest area of a minimal surface (Q2499330)

In this paper, the authors 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 is given in terms of the diameter of \(M^n\), the second estimate in terms of the filling radius of a manifold, leads to an estimate in terms of the volume of \(M^n\). If \(n=3\), their upper bounds are for the smallest area of a smooth embedded minimal surface. The authors also establish similar upper bounds for the smallest volume of a stationary \(k\)-dimensional integral varifold in a closed Riemannian manifold \(M^n\) with \(H_1(M^n)=\dots H_{k-1}(M^n)=\{0\}, (k>2)\).