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.

To appear in Geometric and Functional Analysis

Cited by in corpus (1)

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