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.
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The authors thank Tübingen University and the Mathematisches Forschungsinstitut Oberwolfach where part of this work was carried out. The first author was supported in part by the National Science Foundation under Grant DMS-1505724.
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Brendle, S., Hung, PK. Area Bounds for Minimal Surfaces that Pass Through a Prescribed Point in a Ball. Geom. Funct. Anal. 27, 235–239 (2017). https://doi.org/10.1007/s00039-017-0399-6
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DOI: https://doi.org/10.1007/s00039-017-0399-6