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The degree condition in Llarull’s theorem on scalar curvature rigidity
[La condition sur le degré dans le théorème de Llarull concernant la rigidité de la courbure scalaire]
Christian Bär1 ; Rudolf Zeidler1
1Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 243-252

Llarull’s scalar curvature rigidity theorem states that a $1$-Lipschitz map $f\colon M\rightarrow \mathbb{S}^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\operatorname{scal}\ge n(n-1)$ to the standard sphere $\mathbb{S}^n$ is an isometry if the degree of $f$ is nonzero. We investigate if one can replace the condition $\deg (f)\ne 0$ by the weaker condition that $f$ is surjective. The answer turns out to be “no” for $n\ge 3$ but “yes” for $n=2$. If we replace the scalar curvature by Ricci curvature, the answer is “yes”in all dimensions.

Le théorème de rigidité de Llarull sur la courbure scalaire énonce qu’une application $1$-lipschitzienne $f\colon M\rightarrow \mathbb{S}^n$, d’une variété riemannienne fermée et connexe $M$, dotée d’une courbure scalaire $\operatorname{scal}\ge n(n-1)$, vers la sphère standard $\mathbb{S}^n$, est une isométrie si le degré de $f$ est non nul. Nous étudions si l’on peut remplacer la condition $\deg (f)\ne 0$ par la condition plus faible selon laquelle $f$ est surjective. Il s’avère que la réponse est « non » pour $n\ge 3$, mais « oui » pour $n=2$. Si nous remplaçons la courbure scalaire par la courbure de Ricci, la réponse est « oui » dans toutes les dimensions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.825
Classification : 53C20, 53C24
Keywords: Scalar curvature, Ricci curvature, rigidity, Llarull’s theorem, Lipschitz-volume rigidity
Mots-clés : Courbure scalaire, courbure de Ricci, rigidité, théorème de Llarull, rigidité de Lipschitz-volume

Christian Bär  1   ; Rudolf Zeidler  1

1 Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {The degree condition in {Llarull{\textquoteright}s} theorem on scalar curvature rigidity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {243--252},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.825},
     language = {en},
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Christian Bär; Rudolf Zeidler. The degree condition in Llarull’s theorem on scalar curvature rigidity. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 243-252. doi: 10.5802/crmath.825

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