Area bounds for minimal surfaces that pass through a prescribed point in a ball

Abstract

Let $\Sigma$ be a $k$-dimensional minimal submanifold in the $n$-dimensional unit ball $B^n$ which passes through a point $y \in B^n$ and satisfies $\partial \Sigma \subset \partial B^n$. We show that the $k$-dimensional area of $\Sigma$ is bounded from below by $|B^k| \, (1-|y|^2)^{\frac{k}{2}}$. This settles a question left open by the work of Alexander and Osserman in 1973.