On Meeks' conjecture on disjoint minimal graphs

We study disjoint solution pairs $(u,D)$ to the minimal surface equation in ${\mathbb{R}}^n$, with $u=0$ on $\mathrm{\partial }D\mathrm{.}$ A conjecture of W. Meeks states that at most two such pairs can exist in ${\mathbb{R}}^n\mathrm{.}$ We confirm the conjecture in two complementary cases. For $n\in \{2,3,4,5,6\},$ we prove it under a superlinearity assumption. In dimension $n=2,$ we establish the result under a suitable sublinearity assumption. We also derive estimates on the asymptotic angle of domains supporting nontrivial solutions for $n\ge 3.$