Geometry of three-dimensional manifolds with scalar curvature lower bound

Abstract

This paper is a continuation of our previous work concerning three-dimensional complete manifolds with scalar curvature bounded from below. One of the purposes is to improve a sharp comparison theorem for the bottom spectrum by removing a volume assumption on unit balls. Another purpose is to derive geometric information when the scalar curvature is assumed to be bounded from below by a positive constant. When the Ricci curvature is asymptotically nonnegative, it is shown that such manifolds must be parabolic and that the lower bound of the scalar curvature is explicitly bounded by the volume of unit balls. In particular, in the case that the Ricci curvature is nonnegative, this implies that the volume of the manifold must grow linearly, which answers a question of Gromov for dimension three.

Topological Characterization of Contractible 3-Manifolds with Positive Scalar Curvature

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In this note, we prove an effective linear volume growth for complete three-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. This recovers the results obtained by Munteanu-Wang [7]. Our method builds upon recent work by Chodosh-Li-Stryker [4], which utilizes the technique of µ-bubbles and the almostsplitting theorem by Cheeger-Colding.

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“…An alternative method has been developed by Munteanu-Wang [7], which relies on analysis of level sets of harmonic functions. We refer some other related works for this problem in higher dimensions [8], [10].…”

Monotonicity of the $p$-Green functions

“…This provides a new pathway to study complete 3-dimensional manifolds with nonnegative scalar curvature R M ≥ 0. The method is far-reaching and has been proven to be successful in view of the recent progresses in [5,6,10,35,36]. In particular, Bray-Kazaras-Khuri-Stern [6] introduced a new formula relating certain integral of the harmonic functions to the ADM mass of an asymptotically flat manifold and give an alternative proof to the positive mass theorem in dimension 3 (see also [5] for a proof using monotonicity of harmonic function).…”

“…In particular, Bray-Kazaras-Khuri-Stern [6] introduced a new formula relating certain integral of the harmonic functions to the ADM mass of an asymptotically flat manifold and give an alternative proof to the positive mass theorem in dimension 3 (see also [5] for a proof using monotonicity of harmonic function). Motivated by the results in [11,13], Munteanu-Wang [35,36] investigated the quantity…”

On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds

“…More recently, the work of O. Munteanu and J. Wang [MW23,MW22] implies that if the closed 3-manifold (M, g) satisfies R g ≥ −6, then…”