Low-conformal-volume isotopy conjecture for minimal surfaces in hyperbolic three-space

Let $\Gamma\subset \mathbb{S}^{2}$ be a closed curve, and let $\lambda_c[\Gamma]$ denote its conformal volume. Let $\lambda[\mathbb{S}^{1}\times\mathbb{R}]$ be the entropy of the cylinder, and let $\Sigma_1,\Sigma_2\subset\mathbb{H}^{3}$ be minimal surfaces with $C^1$-regular asymptotic boundaries.

Low-conformal-volume isotopy conjecture. If

$$\lambda_c[\Gamma]\leq\lambda[\mathbb{S}^{1}\times\mathbb{R}]$$

and

$$\partial_\infty\Sigma_1=\partial_\infty\Sigma_2=\Gamma,$$

then $\Sigma_1$ is isotopic to $\Sigma_2$.

The conjecture is motivated by topological uniqueness results for low-entropy self-expanders. Its restriction to curves in $\mathbb{S}^{2}$ reflects the special properties of renormalized area in dimension two; the conjecture itself is not resolved in the source.