PreprintCC BY 4.0arXiv
Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
Abstract
We prove that, in every closed Riemannian three-manifold (M3,g̅), there exists a constant C=C(M,g̅)>0 such that every closed smoothly embedded minimal surface Σ⊂ M satisfies b1(Σ;ℤ2)≤ C(Ind(Σ)+Area(Σ)), establishing Song's additive genus-index-area conjecture. For closed connected orientable two-sided minimal immersions, we prove the universal inequality γ(Σ)≤8Ind(Σ) in positive Ricci curvature and the estimate γ(Σ)≤ C(M,g̅)Ind(Σ) in positive scalar curvature. In the latter case, we also bound both area and total curvature by C(M,g̅)(1+Ind(Σ)).
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BibTeX
@misc{caniato2026topology,
title = {{Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds}},
author = {Riccardo Caniato},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32001},
eprint = {2609.32001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32001},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
AU - Riccardo Caniato
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.32001
UR - https://arxiv.org/abs/2609.32001
AB - We prove that, in every closed Riemannian three-manifold (M^3,g̅), there exists a constant C=C(M,g̅)>0 such that every closed smoothly embedded minimal surface Σ⊂ M satisfies b_1(Σ;ℤ_2)≤ C(Ind(Σ)+Area(Σ)), establishing Song's additive genus-index-area conjecture. For closed connected orientable two-sided minimal immersions, we prove the universal inequality γ(Σ)≤8Ind(Σ) in positive Ricci curvature and the estimate γ(Σ)≤ C(M,g̅)Ind(Σ) in positive scalar curvature. In the latter case, we also bound both area and total curvature by C(M,g̅)(1+Ind(Σ)).
ER -