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:
- If $r=0$, the statement follows directly from the monotonicity formula.
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 $\Sigma$ is topological disc the answer is YES, see answer of Oleg Eroshkin below.
If $r=0$, the statement follows directly from the monotonicity formula.
- The general question is formulated as a conjecture in 1975 --- see comment of Ian Agol.
If $\Sigma$ is topological disc the answer is YES, see answer of Oleg Eroshkin below.
- 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.
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.
