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:

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.)

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.