Scalar curvature rigidity for products of spheres and tori

2025/11/06 by Chow, Tsz-Kiu Aaron · 1 citation
#53C21 #53C24 #53C27 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.04407

Abstract

We prove Llarull-type rigidity for S^n-m×\mathbbT^m (3≤ n≤ 7, 1≤ m≤ n-2). If a closed spin (Mⁿ,g) admits a degree-nonzero map to S^n-m×\mathbbT^m whose spherical projection is area non-increasing, and there exists ψ∈ C^∞(M) with -Δ_Mψ-(1)/(2)|D_Mψ|²+(1)/(2)(R_M-(n-m)(n-m-1))≥0, then (M,g) is isometrically covered by S^n-m×ℝ^m. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: dist(∂_-M,∂₊M)≤ 2π√(n/((n+1)σ)) when R_M≥ (n-m)(n-m-1)+σ. The method combines stable weighted slicing with a spectral Dirac operator argument.

Citations

• The rigidity statement in the Horowitz-Myers conjecture
• Dirac eigenvalues and the hyperspherical radius
• Systolic inequalities and the Horowitz-Myers conjecture
• Llarull type theorems on complete manifolds with positive scalar curvature
• Rigidity results for initial data sets satisfying the dominant energy condition
• Spectral Torical Band Inequalities and Generalizations of the Schoen-Yau Black Hole Existence Theorem
• Rigid comparison geometry for Riemannian bands and open incomplete manifolds
• Rigidity on non-negative intermediate curvature
• Rigidity of Lipschitz map using harmonic map heat flow
• A generalization of Geroch's conjecture
• Lipschitz rigidity for scalar curvature
• Index theory for scalar curvature on manifolds with boundary
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• Width estimate and doubly warped product
• Four Lectures on Scalar Curvature
• Rigidity of Area-Minimizing 2-Spheres in n-Manifolds with Positive Scalar Curvature
• Metric Inequalities with Scalar Curvature
• Positive Scalar Curvature and Minimal Hypersurface Singularities
• Scalar Curvature on Compact Symmetric Spaces
• Rigidity of area-minimizing two-spheres in three-manifolds

Cited by

• A Weighted Llarull Type Theorem and its applications

Related

• Scalar curvature rigidity of spheres with subsets removed and L^∞ metrics
• Spectral Flow, Llarull's Rigidity Theorem in Odd Dimensions and its Generalization
• Spectral constant rigidity of warped product metrics
• Scalar curvature rigidity of domains in a warped product
• Stability of Llarull's theorem in all dimensions