Positive Scalar Curvature and Volume Growth

Abstract

For a complete Riemannian manifold with nonnegative Ricci curvature, we prove two sharp volume growth order estimates, thereby resolve a conjecture of Gromov in 1986. There first is that a uniform deficit in the volume of unit balls, an analog of positive macroscopic scalar curvature, forces codimension one volume growth, and the second one is that a uniformly positive scalar curvature lower bound forces codimension two growth known as the codimension two volume growth conjecture.

Disclosure

“ume growth of a geodesic ball. To apply this idea to all geodesics, a geodesic flow argument and a pairing are needed, which are the contents of Proposition 5.2 and 5.3. Acknowledgements. X.Z is supported by AMS Travel Fund. AI disclosure. Generative AI tools, more explicitly, ChatGPT 5.6 Sol Ultra and Codex, assisted with proof exploration, organization, and drafting. Essential ideas were generated by AI. The authors reviewed, edited, and remain responsible for all theorem statements, pr”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:15:26.983328Z

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