Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature and Fundamental Group Decay

2026/06/28 by Qiaochu Ma, Guoliang Yu
#math.DG #math.KT

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Abstract

A conjecture of Gromov predicts that every closed oriented manifold admitting a metric of positive scalar curvature has vanishing simplicial volume. In this paper, we prove this conjecture under a mild decay condition on the fundamental group. As a geometric application, we show that, in every dimension, there exists a universal negative constant such that any closed oriented manifold whose scalar curvature is bounded below by a normalized positive constant and whose Ricci curvature is bounded below by the universal constant has vanishing simplicial volume.

Citations

• A new proof of the positive energy theorem
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• On the proof of the positive mass conjecture in general relativity

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