Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions

Abstract: The main result of this article is a Llarull-type rigidity statement for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions. The even dimensional analog was proven in an earlier work together with Simone Cecchini, Bernhard Hanke and Thomas Schick using index theory and the analysis of abstract cone operators, which applies to Dirac operators associated with generalized cone metrics. We will extend the analysis of abstract cone operators, apply it to twisted Dirac operators on singular manifolds and combine it with a spectral flow argument to prove the main result.

• The paper establishes a Llarull-type rigidity theorem showing that a Lipschitz, area non-increasing map forces isometry away from cone singularities on odd-dimensional spin manifolds.
• It employs advanced index theory, abstract cone operator analysis, and twisted Dirac operators to connect analytic and topological properties of the manifold.
• The spectral flow, linked to the mapping degree, demonstrates that the manifold is diffeomorphic to a punctured sphere, reinforcing scalar curvature rigidity in singular settings.

Lipschitz Rigidity for Scalar Curvature on Singular Manifolds in Odd Dimensions

Introduction

The paper "Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions" (2605.00151) establishes a Llarull-type rigidity theorem for scalar curvature on odd-dimensional Riemannian spin manifolds with cone-like singularities. This result generalizes scalar curvature rigidity in the context of singular spaces and addresses a gap left open in the literature for odd dimensions, extending previous work in the even-dimensional setting [CHSS]. The approach employs advanced index theory, the spectral theory of abstract cone operators, twisted Dirac operators, and spectral flow techniques, building upon developments in scalar curvature rigidity and the interplay between geometric analysis and global differential geometry.

Mathematical Framework and Main Result

The analysis focuses on compact Riemannian spin manifolds $(N,G)$ of dimension $n+1$ (with $n$ even), allowing cone-like singularities. The notion of cone-like singularity is formalized via generalized cone metrics, for which a precise analytic structure is described using families of Riemannian metrics $\{g_r\}_{r \in (0,\vartheta)}$ with detailed regularity and asymptotic behavior at $r \to 0$. The definition includes control conditions on derivatives to ensure the cone structure is analytically tractable.

The principal theorem asserts that if $(N,G)$ admits a spin structure, has scalar curvature satisfying $\scal_G \geq (n+1)n$, and supports a Lipschitz, area non-increasing map $$f: (N,G) \to (\mathbb{S}^{n+1}, g_{\mathbb{S}^{n+1}})$$ of non-zero degree, then $f$ is a Riemannian isometry away from the cone points, and $N$ is diffeomorphic to a punctured sphere. This generalizes the rigidity phenomenon of Llarull’s theorem [Llarull], removing smoothness and compactness assumptions, and allowing singular geometry along finite cone points.

Abstract Cone Operators and Dirac Analysis

A substantial part of the paper is devoted to the extension and analysis of abstract cone operators, building on foundational work by Br\"uning and Seeley [BrueningSeeley87, Bruening90]. Cone operators are constructed as densely defined, closable operators on Hilbert spaces capturing the singular behavior near cone points, with a specific product structure: $n+1$0. Here, $n+1$1 is the link operator encoding geometry of the cross-sectional manifold, and $n+1$2 is a perturbation controlling the deviation from the straight cone case.

The paper outlines spectral gap conditions, compactness criteria for domains, and self-adjointness results for these operators. Of particular technical importance is the geometric Witt condition—ensuring the spectrum of $n+1$3 avoids a symmetric interval around zero—which is critical for self-adjointness and Fredholm theory.

By showing that Dirac operators twisted with Lipschitz pullback bundles on such singular manifolds are indeed self-adjoint abstract cone operators with discrete spectrum and finite multiplicities, the analytic foundation for index theory in the singular setting is provided. The construction involves careful local trivializations, boundedness controls, and the use of parallel transport and heat kernel techniques.

Spectral Flow and Deformation Arguments

A central technical tool in the proof is the spectral flow of families of self-adjoint Fredholm operators, continuous in the gap topology [BoosLeschPhillips]. The Dirac operator on $n+1$4 is deformed via families in three aspects: the twist connection, the comparison map, and the cone metric. The gap topology continuity is ensured via uniform control on domains and equivalence of graph norms.

The paper implements a deformation strategy whereby the twisted Dirac operator is homotoped to a situation in which the comparison map is constant near singularities and the metric is a straight cone. This enables reduction to a closed case previously treated in [Baer_2024, LiSuWang], where the spectral flow is computed explicitly using Getzler’s formula [Getzler], involving the heat kernel and trace computation:

$n+1$5

Strong numerical results include the identification of the spectral flow as the mapping degree, which is a bold claim in this context. The spectral gap for the Dirac operator is also quantified in terms of scalar curvature lower bounds.

Proof of Rigidity and Implications

Combining the analytic and topological tools, the argument shows that non-invertibility of the Dirac operator (via nonzero spectral flow) forces the existence of harmonic spinors. Scalar curvature rigidity is deduced from the integrated Schrödinger–Lichnerowicz formula, yielding equality precisely for local isometries. The result is sharpened by a global argument (invoking the Myers–Steenrod theorem), showing that the comparison map is globally a Riemannian isometry away from cone points, and the manifold is diffeomorphic to a punctured sphere.

The theoretical implication is a striking extension of scalar curvature rigidity to singular spaces, connecting spin geometry, index theory, and metric measure geometry. This demonstrates robustness of geometric analytic methods in the presence of singularities, opening pathways for further study of scalar curvature and rigidity phenomena in non-smooth, non-compact, or singular geometric contexts.

Practically, these results support new avenues for the classification of spaces with positive scalar curvature and controlled singularity, and may inform future developments in curvature-sensitive analysis and study of metric geometry under Lipschitz transformations.

Future Directions

Potential developments include relaxation of regularity conditions, extension to iterated or more general singularity types, and exploration of the quantitative aspects of rigidity via spectral invariants. Connections with synthetic curvature bounds (e.g., in $n+1$6-metrics [ChuLeeZhu24]) and other index-theoretic settings are natural next steps. Further generalizations to other geometric operators and analytic invariants in singular or collapsed space scenarios remain an open challenge, as does the investigation of broader topological consequences for moduli spaces of metrics with lower scalar curvature bounds.

Conclusion

The paper rigorously establishes scalar curvature rigidity for odd-dimensional spin manifolds with cone-like singularities, employing a combination of abstract cone operator theory, Dirac operator analysis, gap topology arguments, and spectral flow computations. The results generalize Llarull-type rigidity to singular manifolds, underscoring the depth and flexibility of index-theoretic and analytic methods in geometric analysis. The implications span both foundational geometric topology and practical metric geometry, setting a benchmark for further research in scalar curvature and rigidity phenomena in singular spaces.