Index estimates for complete noncompact free boundary minimal surfaces
Abstract
Let Ω⊂ℝ3 be an unbounded domain with smooth boundary and let Σ be a complete, noncompact, orientable, immersed free boundary minimal surface in Ω, with compact boundary ∂ Σ⊂∂ Ω and finite Morse index. We prove that if the mean curvature of ∂ Ω satisfies H∂ Ω≥ 0 along ∂ Σ and H∂Ω>0 at some point of ∂Σ, then Ind(Σ) ≥ 1/3(2g+k+2∑j=1r(dj+1)−2), where g is the genus of Σ, k is the number of connected components of ∂Σ, r is the number of ends of Σ and d1,…,dr are their respective multiplicities. When H∂Ω is only assumed to be nonnegative we obtain Ind(Σ)≥(2g+k−2)/3, with the sharp bound (2g+k−1)/3 under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh--Máximo, with weighted L2 spaces, adapted to the free boundary setting in the spirit of Ambrozio--Carlotto--Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights ρ, we prove that this dimension is 2g+k−1+2∑jNj−ε, where Nj is the maximal order of pole allowed by ρ at the j-th puncture and ε∈{0,1}. For the weight of Chodosh--Máximo one has Nj=dj+1.
Related
- Morse index, topology, and ends of minimal surfaces with noncompact free boundary
- Sub-Laplacian Comparison Theorem and Gradient Estimates on Complete Noncompact Contact Riemannian Manifolds
- Nonlocal Approximation of Minimal Surfaces: Optimal Estimates From Stability
- Volumetric density estimates for nonlocal minimal surfaces
- Free boundary minimal surfaces with connected boundary and arbitrary genus
BibTeX
@misc{cavalcante2026index,
title = {{Index estimates for complete noncompact free boundary minimal surfaces}},
author = {Marcos P. Cavalcante and Abraao Mendes and Ian R. dos Santos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27651},
eprint = {2609.27651},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27651},
}
RIS
TY - UNPB
TI - Index estimates for complete noncompact free boundary minimal surfaces
AU - Marcos P. Cavalcante
AU - Abraao Mendes
AU - Ian R. dos Santos
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27651
UR - https://arxiv.org/abs/2609.27651
AB - Let Ω⊂ℝ^3 be an unbounded domain with smooth boundary and let Σ be a complete, noncompact, orientable, immersed free boundary minimal surface in Ω, with compact boundary ∂ Σ⊂∂ Ω and finite Morse index. We prove that if the mean curvature of ∂ Ω satisfies H_∂ Ω≥ 0 along ∂ Σ and H_∂Ω>0 at some point of ∂Σ, then Ind(Σ) ≥ 1/3(2g+k+2∑_j=1^r(d_j+1)−2), where g is the genus of Σ, k is the number of connected components of ∂Σ, r is the number of ends of Σ and d_1,…,d_r are their respective multiplicities. When H_∂Ω is only assumed to be nonnegative we obtain Ind(Σ)≥(2g+k−2)/3, with the sharp bound (2g+k−1)/3 under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh--Máximo, with weighted L^2 spaces, adapted to the free boundary setting in the spirit of Ambrozio--Carlotto--Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights ρ, we prove that this dimension is 2g+k−1+2∑_jN_j−ε, where N_j is the maximal order of pole allowed by ρ at the j-th puncture and ε∈{0,1}. For the weight of Chodosh--Máximo one has N_j=d_j+1.
ER -