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arXiv · 2610.20508

On Meeks' conjecture on disjoint minimal graphs

Angelo Benedetti·Barbara Nelli
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Abstract

We study disjoint solution pairs $(u,D)$ to the minimal surface equation in $\mathbb{R}^n$, with $u=0$ on $\partial D.$ A conjecture of W. Meeks states that at most two such pairs can exist in $\mathbb{R}^n.$ 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 $n\geq 3.$

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Angelo Benedetti, Barbara Nelli. 2026-10-14. On Meeks' conjecture on disjoint minimal graphs. https://arxiv.org/abs/2610.20508

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