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This research aims to investigate whether the degree condition in Llarull’s theorem can be relaxed to surjectivity in certain cases.

The degree condition in Llarull’s theorem on scalar curvature rigidity

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This research aims to investigate whether the degree condition in Llarull’s theorem can be relaxed to surjectivity in certain cases.
Analyzing scalar curvature conditions in closed connected Riemannian spin manifolds.
Comparing conditions under which isometries hold for different dimensions.
Examining the results for both scalar and Ricci curvature.
The degree condition cannot be replaced by surjectivity for manifolds with dimension n ≥ 3.
For the case of dimension n = 2, surjectivity suffices for isometry.
When Ricci curvature is considered instead, surjectivity maintains isometric properties in all dimensions.

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Analysis of degree conditions for isometries in Riemannian manifolds, revealing important conditions for rigidity.

Bär et al. (2026) studied this question.

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