The degree condition in Llarull’s theorem on scalar curvature rigidity
Abstract
Llarull’s scalar curvature rigidity theorem states that a 1 -Lipschitz map f : M → 𝕊 n from a closed connected Riemannian spin manifold M with scalar curvature scal ≥ n ( n - 1 ) to the standard sphere 𝕊 n is an isometry if the degree of f is nonzero. We investigate if one can replace the condition deg ( f ) ≠ 0 by the weaker condition that f is surjective. The answer turns out to be “no” for n ≥ 3 but “yes” for n = 2 . If we replace the scalar curvature by Ricci curvature, the answer is “yes”in all dimensions.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Nonlinear Partial Differential Equations
Advanced Differential Geometry Research
Publication details
- Journal / source
- Comptes Rendus Mathématique
- Volume
- 364
- Issue
- G1
- Pages
- 243–252
- Publication date
- 17 March 2026
- Language
- EN