Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.29853

Sharp monotonicity for closed three-manifolds with positive scalar curvature

Cosmin Manea
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove two sharp monotonicity formulae for the natural Green's function of a closed three-manifold with positive scalar curvature, analogous to the Munteanu-Wang theorem for non-negative scalar curvature. We also show that both results may be interpreted as the monotonicity of a suitably defined "enclosed mass" of the sublevel sets of Green's function in the corresponding "conformal" blow-up of the manifold.

Explore related subjects

math.DGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cosmin Manea. 2026-08-30. Sharp monotonicity for closed three-manifolds with positive scalar curvature. https://arxiv.org/abs/2608.29853

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗
arXiv2026

Isoparametric foliations and bounded geometry

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.

math.DG↗
arXiv2026

Minimal foliations, codimension-one stable norms, and a question of Bangert

We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension $n\ge3$, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On $\mathbb T^3$, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on $\mathbb T^3$ admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.

math.DG↗