Marked Length Spectrum Rigidity of Simple, Thick Negatively Curved Two-Dimensional P-Manifolds

Core Concepts

Simple, thick negatively curved two-dimensional P-manifolds are marked length spectrum rigid, meaning their piecewise negatively curved Riemannian metrics are determined (up to isotopy) by the lengths of their closed geodesics.

Abstract

• Bibliographic Information: Wu, Y. (2024). Marked Length Spectrum Rigidity for Surface Amalgams [Preprint]. arXiv:2310.09968v2

• Research Objective: This paper investigates the marked length spectrum rigidity of simple, thick negatively curved two-dimensional P-manifolds, aiming to determine if their metrics are uniquely determined by the lengths of their closed geodesics.

• Methodology: The author generalizes the techniques used by Croke and Otal in their proof of marked length spectrum rigidity for negatively curved surfaces. The proof leverages the properties of CAT(-1) spaces, Gromov hyperbolic spaces, and the ergodicity of the geodesic flow map on P-manifolds. The author also utilizes separability properties of fundamental groups of surface amalgams and their relation to non-positively curved cube complexes to extend the results to a broader class of P-manifolds.

• Key Findings: The paper proves that if two simple, thick negatively curved P-manifolds, equipped with specific piecewise Riemannian metrics, have the same marked length spectrum, then there exists an isometry between them that is isotopic to the identity. This finding significantly extends the understanding of marked length spectrum rigidity beyond surfaces to a class of spaces constructed by gluing surfaces together.

• Main Conclusions: The research establishes a strong rigidity result for a large class of surface amalgams, demonstrating that their geometry is highly constrained by their marked length spectrum. This result has implications for the study of geometric group theory, particularly in understanding the relationship between the algebraic and geometric properties of groups.

• Significance: This work contributes significantly to the field of geometric rigidity, expanding the known cases of marked length spectrum rigidity beyond manifolds. It also highlights the fruitful interplay between geometric group theory, hyperbolic geometry, and dynamical systems in studying the rigidity phenomena of non-positively curved spaces.

• Limitations and Future Research: The current work focuses on simple, thick P-manifolds with specific smoothness conditions on their metrics. Future research could explore the possibility of relaxing these conditions or investigating marked length spectrum rigidity for higher-dimensional P-manifolds. Additionally, exploring the implications of this rigidity result for the study of mapping class groups of P-manifolds could be a promising research direction.

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Marked Length Spectrum Rigidity for Surface Amalgams

by Yandi Wu at arxiv.org 11-19-2024

https://arxiv.org/pdf/2310.09968.pdf

Deeper Inquiries

Can the smoothness conditions imposed on the metrics of the P-manifolds in this study be relaxed while still ensuring marked length spectrum rigidity?

This is a nuanced and insightful question that gets at the heart of the technical challenges in extending rigidity results to broader classes of metrics. Here's a breakdown of the answer, incorporating relevant terminology:
Current Restrictions and Why They Exist:

The paper focuses on metrics in the class M≤, which are piecewise Riemannian with negative sectional curvature bounded above by -1. Crucially, they demand a smooth transition across gluing curves (Condition 3 in the definition of M≤).
This smoothness is not merely a technical convenience. Classical proofs of marked length spectrum rigidity for surfaces (Croke, Otal) heavily rely on the well-defined unit tangent bundle and the ergodicity of the geodesic flow on this bundle. Smoothness guarantees this.
Potential for Relaxation:

Cone Singularities: The author correctly points out that finitely many cone singularities within chambers could be handled using existing techniques (Hersonksy-Paulin, Constantine). This is because the singular points are isolated, and the flow remains well-behaved elsewhere.
Non-Smooth Gluing: Relaxing Condition 3 is far more challenging. The generalized unit tangent bundle, while existing, becomes more complex. Proving ergodicity of the flow requires different tools. Directly adapting Otal's proof might not be possible.
Approaches and Open Questions:

Alternative Ergodicity Proofs: Could one establish ergodicity of a suitable flow on the generalized tangent bundle for a broader class of metrics? This would be a significant advancement.
Weakening the Conclusion: Perhaps full marked length spectrum rigidity is too strong a requirement without smoothness. Could one obtain weaker forms of rigidity (e.g., finite-to-one maps, or rigidity up to some controlled deformation of the metric)?
New Invariants: Are there other geometric invariants that capture the metric information lost when smoothness is relaxed, which could supplement the marked length spectrum?
In summary, relaxing smoothness at the gluing curves is a difficult open problem. It likely requires new ideas about studying geodesic flows in singular spaces or finding alternative geometric invariants.

Could there be alternative geometric or topological invariants, besides the marked length spectrum, that also determine the metric of a P-manifold up to isotopy?

This is an excellent question that probes the broader landscape of rigidity phenomena. Here are some possibilities and considerations:
Candidates for Alternative Invariants:

Marked Boundary Distance Function: For negatively curved surfaces with boundary, this function assigns to each pair of boundary points the length of the geodesic connecting them. Guillarmou-Mazzucchelli have shown rigidity results using this for surfaces. Generalizing to P-manifolds might be fruitful.

Spectrum of the Laplacian: The eigenvalues of the Laplace-Beltrami operator on a Riemannian manifold encode geometric information. While not as directly linked to geodesics, spectral rigidity results exist for some spaces. Investigating this for P-manifolds could be interesting.

Dynamical Invariants of the Geodesic Flow: Beyond ergodicity, more refined properties of the geodesic flow (e.g., mixing rates, entropy, topological entropy) might hold the key to rigidity for less regular metrics.

Combinatorial Invariants of the Universal Cover: The structure of the universal cover ẽX as a tree of spaces, along with data about how chambers are glued, could potentially be used to construct invariants. This would require a deeper understanding of the interplay between the combinatorial and geometric aspects of ẽX.

Challenges and Considerations:

Sensitivity to Smoothness: Many geometric invariants, like the Laplacian spectrum, are highly sensitive to the regularity of the metric. Finding invariants robust to non-smoothness is a major challenge.
Computability and Practicality: An ideal invariant should be computable (at least in principle) and provide a practical way to distinguish non-isometric metrics.
In conclusion, exploring alternative invariants is a promising direction. It requires a delicate balance between finding invariants that are both robust to the irregularities of P-manifolds and powerful enough to capture their geometry.

How does the concept of marked length spectrum rigidity relate to the study of the dynamics of geodesic flows on more general non-positively curved spaces beyond P-manifolds?

This question highlights the deep connections between rigidity phenomena, geometric invariants, and the dynamics of geodesic flows. Here's an exploration of the relationship:
Marked Length Spectrum and Geodesic Flows:

Fundamental Link: The marked length spectrum is intimately tied to the lengths of closed geodesics, which are precisely the periodic orbits of the geodesic flow. Rigidity implies that the lengths of these orbits, a purely dynamical notion, determine the geometry of the space.

Ergodicity as a Bridge: As seen in the paper, ergodicity of the geodesic flow is a crucial ingredient in proving marked length spectrum rigidity. Ergodicity implies that the flow "explores" the space uniformly, allowing one to relate local information (cross ratios) to global information (lengths of closed geodesics).
Beyond P-Manifolds: General Non-Positively Curved Spaces:

Difficulties and Open Questions: For general non-positively curved spaces, the situation is far more complex.

Geodesic flows might not be ergodic.
Even if ergodic, the link between dynamics and geometry might be weaker.
Singularities and other irregularities pose significant challenges.

Areas of Active Research:

Rank Rigidity: In higher-rank symmetric spaces, the marked length spectrum alone is not rigid. Understanding what additional data is needed is an active area of research.
Geodesic Flows on Non-Manifolds: Studying geodesic flows on cube complexes, CAT(0) spaces, and other singular spaces is an active field with connections to geometric group theory.
Entropy Rigidity: Results relating the entropy of the geodesic flow to geometric invariants like volume offer another perspective on rigidity.
In essence, marked length spectrum rigidity exemplifies a powerful interplay between geometry and dynamics. While well-understood for negatively curved surfaces and some generalizations like P-manifolds, extending these ideas to more general non-positively curved spaces requires grappling with the complexities of their geodesic flows and finding new ways to relate dynamics to geometry.