Some rigidity theorems for spectral curvature bounds

Rigidity Theorems for Spectral Curvature Bounds

Introduction

This work systematically extends classical rigidity phenomena for scalar curvature in Riemannian geometry to the setting of spectral curvature, defined via eigenvalue-type elliptic operators involving both metric and curvature data. Employing the warped $\mu$-bubble formalism—a generalization of weighted minimal hypersurfaces—the authors establish new classification and rigidity results for stable weighted minimal hypersurfaces under spectral scalar and Ricci curvature bounds. Significant consequences include extensions of the Geroch conjecture and band width estimates in the spectral regime, as well as splitting results for manifolds with arbitrary ends.

Spectral Curvature Framework and Weighted Minimal Hypersurfaces

Spectral curvature considers operators of the form $-\gamma \Delta_g u + \frac{1}{2} R_g u$ (and analogues involving Ricci curvature), where, crucially, these “curvatures” depend not only on the geometry but also on a positive function $u$ and parameter $\gamma$. This paradigm interpolates between scalar curvature rigidity (standard case $u=1$, $\gamma=0$) and stability questions for certain non-linear PDEs, linking to prominent conjectures such as Bernstein, aspherical, and Geroch.

To capture the variational essence of hypersurface stability in this setting, the $u^\gamma$-weighted area functional is employed, whose critical points are the so-called weighted minimal hypersurfaces or, more generally, the warped $\mu$-bubbles. The second variation formula is thoroughly derived and then reformulated to assess stability and rigidity using tools from analysis and conformal geometry.

Classification of Stable Weighted Minimal Hypersurfaces

A principal result is a structural classification theorem for stable, complete, oriented, weighted minimal surfaces in 3-manifolds under non-negative spectral scalar curvature with parameter $0 \leq \gamma < 4$:

• If the hypersurface $\Sigma$ is compact, it must be either a sphere or a torus, and, in the latter case, flatness and local splitting of the ambient manifold are established if area-minimizing.
• If $-\gamma \Delta_g u + \frac{1}{2} R_g u$0 is non-compact and $-\gamma \Delta_g u + \frac{1}{2} R_g u$1, it is conformally equivalent to either $-\gamma \Delta_g u + \frac{1}{2} R_g u$2 or a cylinder, with a flat metric in the cylindrical case.

The analysis here hinges on a technical rewrite of the second variation formula (using elliptic operator spectral properties and the Gauss-Bonnet theorem), and leverages delicate volume growth arguments for non-compact cases. Notably, a subtle dependence on the parameter range for $-\gamma \Delta_g u + \frac{1}{2} R_g u$3 is required, with phenomena failing for larger $-\gamma \Delta_g u + \frac{1}{2} R_g u$4 unless additional assumptions are imposed.

Analogous statements are established for spectral Ricci curvature, including cases involving $-\gamma \Delta_g u + \frac{1}{2} R_g u$5, further expanding the rigidity landscape.

Band Width Estimates and Quantitative Rigidity

Band width—the distance between the two boundaries of a Riemannian band $-\gamma \Delta_g u + \frac{1}{2} R_g u$6—provides a quantitative measure constraining geometry via curvature bounds. The article leverages warped $-\gamma \Delta_g u + \frac{1}{2} R_g u$7-bubbles with prescribed mean curvature to extend Gromov’s band width techniques to spectral scalar and Ricci curvature settings.

General theorems are established bounding the band width in terms of parameters dictated by spectral curvature lower bounds and weighted mean curvature data on boundaries. In the equality case, rigidity identifies the metric as a specific (doubly) warped or warped product model with explicit warping factor ($-\gamma \Delta_g u + \frac{1}{2} R_g u$8 for suitable $-\gamma \Delta_g u + \frac{1}{2} R_g u$9 depending on curvature and $u$0).

A noteworthy feature is the flexibility to handle $u$1, in the critical ODEs governing the model geometry, which does not occur in the non-spectral ($u$2) regime.

For Ricci spectral curvature, the authors derive a band width bound that strengthens the spectral Bonnet-Myers theorem, allowing for a precise rigidity description in the equality case and demonstrating that certain model manifolds are, up to isometry, the only possible optimizers under the specified spectral bounds.

Splitting Theorems and Extensions of Classical Conjectures

Employing the aforementioned band width results, the authors prove spectral analogues of celebrated splitting theorems. Key among these is a spectral version of the Geroch conjecture: the connected sum $u$3 (where $u$4 is the torus and $u$5 an arbitrary $u$6-manifold, $u$7) cannot carry a complete metric of spectral positive scalar curvature for $u$8. This generalizes deep topological constraints on manifolds admitting metrics with positive scalar curvature to the spectral setting.

Further, for 3-manifolds with non-negative spectral scalar or Ricci curvature and under certain topological conditions (e.g., existence of a properly embedded cylindrical, absolutely weighted area-minimizing surface), global splitting and flatness are deduced, extending results such as the Milnor and Cheeger-Gromoll splitting theorems to the spectral context.

Non-trivial cases, such as manifolds with arbitrary ends or universal covers, are handled via fine analysis of weighted area-minimizing surfaces, PDE deformations of the ambient metric, and detailed study of the variational structure, borrowing techniques from both geometric analysis and global Riemannian geometry.

Technical Contributions and Analytical Developments

A significant methodological advance is the systematic use of the warped $u$9-bubble machinery. This allows both the recovery of classical rigidity phenomena in new settings and the formulation of sharp inequalities (such as band width bounds) in the spectral regime.

The paper derives detailed second variation and stability formulas, adapted for the presence of both the spectral parameter $\gamma$0 and the positive weight function $\gamma$1. These expressions are then reformulated to reveal the underlying conformal invariance and to enable applications of elliptic theory, volume growth estimates, and stability/instability criteria.

Generalizations to operators involving additional gradient terms, such as $\gamma$2, are briefly discussed, connecting the spectral curvature framework to Perelman's scalar curvature (arising in Ricci flow) and further weighted curvatures studied in recent geometric analysis.

Implications and Perspectives

The collection of rigidity theorems for spectral curvature bounds established herein has several important theoretical implications:

• Bridging Scalar and Spectral Curvature Rigidity: The results demonstrate that spectral curvature, despite its analytic complexity, retains a deep connection to topological rigidity phenomena in Riemannian geometry.
• Sharper Quantitative Geometry: The band width estimates and their rigidity cases provide tools for controlling the geometry of manifolds (and bands) with spectral curvature bounds, opening avenues for quantitative comparison geometry in the weighted and spectral settings.
• Topological Constraints: The extension of conjectures known for scalar curvature (e.g., Geroch, Milnor) to the spectral setting further elucidates the intricate interplay between geometric analysis (stability, PDE) and manifold topology.
• Applicability to Weighted and Bakry-Émery Ricci Curvature: The developed techniques are broadly adaptable to settings involving more general weights and measures, suggesting deep connections to comparison geometry for manifolds with density and to research inspired by Ricci flow and optimal transport.

Practically, these theorems contribute to understanding the moduli of manifolds supporting metrics (and weighted structures) with prescribed spectral curvature properties, with potential connections to General Relativity (through the study of the geometry of spacetime slices), geometric flows, and the analysis of non-linear geometric PDEs.

Conclusion

This article provides a unified treatment of rigidity phenomena for spectral curvature bounds, generalizing classical results from scalar curvature geometry and establishing sharp quantitative and qualitative constraints on manifold structure. The warped $\gamma$3-bubble framework underpins new classification theorems, band width bounds, and splitting results, enriching both the analysis and geometry of weighted Riemannian manifolds. The results have far-reaching implications for geometric analysis, comparison geometry, and the study of curvature and topology in both compact and non-compact settings.