Paper · Sep 17, 2026

Total scalar curvature under a curvature operator lower bound

math.DGarXiv:2609.19851

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Abstract

Let $(M^{n}, g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n \geq 3$ , with curvature operator at least that of the unit sphere. We prove that $\int_{M} scal (x) d Vol_{x} \leq n (n - 1) ω_{n},$ where $ω_{n}$ is the volume of the unit $n$ -sphere. Equality holds if and only if $(M, g)$ is isometric to the unit round sphere. In fact, we obtain a stronger bound containing $Vol (M, g)$ . In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.

Comments: 11 pages; comments welcome

Ricci solitons Scalar curvature

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

Total scalar curvature under a curvature operator lower bound

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