Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space
and $r$ is the distance from $\Sigma$ to the center of the ball.
Is it true that
$$\mathop{\rm area} \Sigma\ge \pi\cdot(1-r^2).$$
Comments:
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The problem is solved in all dimensions and codimension, see "Area bounds for minimal…" by Brendle and Hung in 2016. (Thanks Rbega for the ref.)
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If $r=0$, the statement follows directly from the monotonicity formula.
- If $\Sigma$ is topological disc the answer is YES, see answer of Oleg Eroshkin below.
- The general question is formulated as a conjecture in 1975 — see comment of Ian Agol.
- There is an analog in all dimension and codimension for area minimizing surfaces, see Alexander, H.; Hoffman, D.; Osserman, R. Area estimates for submanifolds of Euclidean space. 1974.
Best Answer
This has just been solved (in full generality) by Brendle and Hung using the first variation formula together with a clever (if mysterious) choice of vector field.