As on arXiv
Abstract
Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci flow from rough initial data to reduce the case to smooth metrics and smooth maps so that results by Llarull can be applied.
Comments: 17 pages
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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Short time existence for harmonic map heat flow with time-dependent metrics
Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same author Luen-Fai Tam · Shared terms “map heat”, “heat flow”, “flow coupled”
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Scalar curvature rigidity of the four-dimensional sphere
Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “smooth map”, “zero degree”
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Topics Harmonic maps & DeTurck trick, Weak & singular flows, Scalar curvature · Same author Man-Chun Lee · Shared terms “curvature lower”, “lower bound”
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Topics Stability & uniqueness · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “flow coupled”, “isometry”
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Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “map heat”, “heat flow”
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Topics Harmonic maps & DeTurck trick · Shared terms “map heat”, “heat flow”, “flow coupled”
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Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “map heat”, “heat flow”
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Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “Gromov”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 6C.4The Yamabe Problem and Critical Exponents
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.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.