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Let $${\Sigma}$$Σ be a k-dimensional minimal submanifold in the n-dimensional unit ball Bn which passes through a point $${y \in B^{n}}$$y∈Bn and satisfies $${\partial \Sigma \subset \partial B^{n}}$$∂Σ⊂∂Bn. We show that the k-dimensional area of $${\Sigma}$$Σ is bounded from below by $${{|B^{k}| (1-|y|^{2})}^{\frac{k}{2}}}$$|Bk|(1-|y|2)k2. This settles a question left open by the work of Alexander and Osserman in 1973.

Area Bounds for Minimal Surfaces that Pass Through a Prescribed Point in a Ball