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Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds

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Pith's one-line read Odd-dimensional Llarull rigidity holds for Lipschitz area non-increasing maps and for manifolds with cone-like singularities, proved via spherical suspension and abstract cone operators.

arxiv 2505.14054 v2 pith:HOCXEH3M submitted 2025-05-20 math.DG

Simone Cecchini , Bernhard Hanke , Thomas Schick , Lukas Schoenlinner This is my paper · ORCID
classification math.DG
keywords lipschitzmanifoldsoperatorsabstractcomparisonconecurvaturedirac
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Llarull's theorem says that if a closed manifold with positive scalar curvature admits a 1-Lipschitz map of nonzero degree to the round sphere, then the manifold is isometric to the sphere. This paper proves the same rigidity in odd dimensions for metrics of low regularity (Sobolev W^{1,p}) and for maps that are only area non-increasing, a condition weaker than 1-Lipschitz. The key idea is the spherical suspension: an odd-dimensional manifold M becomes an even-dimensional space S M with cone-like singularities at the poles. The paper develops a general functional-analytic formalism called abstract cone operators to handle Dirac operators on such singular spaces. Using index theory and the Schrödinger-Lichnerowicz formula, the authors show that the suspended map must be an isometry almost everywhere, which then forces the original map to be an isometry. They also prove a second theorem for manifolds that already have cone-like singularities, showing that an area non-increasing Lipschitz map of nonzero degree to the sphere is a smooth Riemannian isometry. The arguments are spinor-geometric and avoid the geometric flow methods used in earlier partial results. This is a step toward understanding how scalar curvature bounds behave under non-smooth comparison maps and on spaces with mild singularities. These results complete the odd-dimensional case of a program initiated by Gromov and extend it to singular geometries.
Extended reading notes

Core claim

The paper's central assertion is Theorem 1.5: if M is a closed smooth connected oriented spin manifold of odd dimension n≥3 with a W^{1,p}-metric g (p>n+1) of distributional scalar curvature at least n(n-1), and f: M→S^n is a Lipschitz map of non-zero degree that is area non-increasing almost everywhere, then f is a metric isometry. The second central assertion, Theorem 1.8, states the same rigidity for compact manifolds with cone-like singularities: if dim N = n+1, n odd ≥3, scal_G ≥ (n+1)n, and f: N→S^{n+1} is Lipschitz, area non-increasing a.e., with non-zero degree, then f is a smooth Riemannian isometry onto S^{n+1} with the cone-tip images removed.

Load-bearing premise

In the proofs of the index formulas (Proposition 3.21, Section 3.3; Proposition 4.15, Section 4.4), the paper assumes the validity of Chou's index theorem for twisted Dirac operators on manifolds with conical singularities, specifically [11, Remark 5.25], for Lipschitz bundles that are trivialized and flat near the tips, together with the deformation-invariance of the Fredholm index under the homotopy of abstract cone operators (Proposition 2.38). If this index-theoretic machinery fails to hold for Lipschitz connections, or if the deformation does not preserve the spectral gap (AC4) uniformly in t, the computation of the index as (-1)^{(n+1)/2} deg(f) χ(S^{n+1}) would be invalid, eliminating the non-zero harmonic spinor that drives the isometry conclusion.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper is a pure mathematical contribution. It introduces no fitted or hand-tuned parameters. It relies on a set of established theorems from index theory, spectral theory, and quasiconformal analysis, which are cited and used as axioms. The abstract cone operator is a new formal construction, not a postulated physical entity.

assumptions (5)
  • domain assumption Index theorem for Dirac operators on manifolds with conical singularities, with flat and trivialized bundle near the tips ([11, Theorem 3.2 and Remark 5.25])
    Invoked in Propositions 3.21 and 4.15 to compute the index of the deformed Dirac operator on the straight cone metric. This is a classical result from the literature, not proved in the paper.
  • standard math Spectral flow invariance for unbounded self-adjoint Fredholm operators ([6, Proposition 2.3])
    Used in Lemma 3.29 (Section 3.4) to equate the spectral flow of D_{g,s} with that of the smooth metric D_{\gamma,s}.
  • domain assumption Quasi-regular maps rigidity theorem (Reshetnyak), as used in [9, Theorem 2.4]
    Invoked at the end of Sections 3.5 and 4.5 to conclude that an orientation-preserving isometry almost everywhere is a metric isometry. This is an external deep result in quasiconformal analysis.
  • domain assumption Integral Schrödinger-Lichnerowicz formula for smooth metrics and trivial bundles ([9, Theorem 5.1])
    The paper extends this formula to Lipschitz bundles and dom(\bar{D}_E) in Theorems 3.18 and 4.12, but the base case is taken from the authors' previous work.
  • standard math Friedrich's spectral gap inequality for Dirac operators on spin manifolds
    Used in Lemma 3.3 and Proposition 3.26 to establish the spectral gap (AC4) for link operators from scalar curvature lower bounds.
invented entities (1)
  • Abstract cone operator
    purpose: A functional-analytic formalism (Definition 2.1) that models Dirac operators on spaces with cone-like singularities, including Lipschitz connections, and supports the construction of parametrices and Fredholm theory.
    This is a newly introduced abstract mathematical object, not an empirical entity. It has no falsifiable experimental handle; its justification is purely internal to the proof.

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Pith. "Pith review of Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds." pith.science (2026). https://pith.science/paper/HOCXEH3M

@misc{pith2026250514054,
  author       = {Pith},
  title        = {Pith review of: Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HOCXEH3M}},
  note         = {Machine review of arXiv:2505.14054}
}
read the original abstract

Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-type singularities and Lipschitz comparison maps to spheres. We use the language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest. Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we generalize a Lipschitz rigidity result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Baer using an upper estimate for the smallest Dirac eigenvalue.

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