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

TL;DR

This paper establishes a lower bound on the area of minimal submanifolds passing through a point in a unit ball, resolving a question posed by Alexander and Osserman in 1973.

Contribution

It provides a sharp area bound for minimal submanifolds passing through a prescribed point in a ball, extending classical results and settling a longstanding open problem.

Findings

The area bound is exactly |B^k| (1 - |y|^2)^{k/2}.

The result applies to k-dimensional minimal submanifolds with boundary on the sphere.

It confirms the conjecture posed by Alexander and Osserman in 1973.

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.