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On Meeks' conjecture on disjoint minimal graphs
Abstract
We study disjoint solution pairs (u,D) to the minimal surface equation in ℝn, with u=0 on ∂ D. A conjecture of W. Meeks states that at most two such pairs can exist in ℝn. We confirm the conjecture in two complementary cases. For n∈{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≥ 3.
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BibTeX
@misc{benedetti2026meeks,
title = {{On Meeks' conjecture on disjoint minimal graphs}},
author = {Angelo Benedetti and Barbara Nelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20508},
eprint = {2610.20508},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20508},
}
RIS
TY - UNPB
TI - On Meeks' conjecture on disjoint minimal graphs
AU - Angelo Benedetti
AU - Barbara Nelli
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20508
UR - https://arxiv.org/abs/2610.20508
AB - We study disjoint solution pairs (u,D) to the minimal surface equation in ℝ^n, with u=0 on ∂ D. A conjecture of W. Meeks states that at most two such pairs can exist in ℝ^n. We confirm the conjecture in two complementary cases. For n∈{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≥ 3.
ER -