Eigenvalue Rigidity in Mathematical Models

• Eigenvalue rigidity is the phenomenon where optimal spectral bounds force underlying structures, such as manifolds and graphs, to match canonical models.
• It employs tools like Bochner identities, variational methods, and random matrix theory to identify extremal configurations with sharp eigenvalue estimates.
• Applications include classifying geometric shapes, ensuring graph rigidity, and quantifying eigenvalue fluctuations in random matrix ensembles.

Eigenvalue rigidity refers to a diverse set of sharp geometric, spectral, and probabilistic phenomena wherein equality or optimal concentration in a spectral inequality forces the ambient structure—be it a manifold, graph, or matrix ensemble—to coincide (often uniquely) with a canonical "model" space. Rigidity theorems typically identify extremal configurations—such as spheres, hypercubes, annuli, or complex projective spaces—by the saturation of bounds for eigenvalues of natural operators: Laplacians, Dirac or Jacobi operators, adjacency matrices, or correlation kernels. The spectrum encodes both geometric structure and stochastic fluctuations, and rigidity results provide an indispensable classification tool linking spectral data to global geometry or combinatorics.

1. Paradigms of Eigenvalue Rigidity: Manifolds and Graphs

In Riemannian geometry, classical rigidity is established through optimal eigenvalue bounds such as the Lichnerowicz–Obata theorem: if a compact n-manifold (M, g) with Ricci curvature Ric ≥ (n–1)g satisfies λ₁(Δ) = n, then (M, g) is isometric to the unit sphere Sⁿ. This connects spectral data (equality in the lower bound for the first Laplacian eigenvalue) with global geometric classification. Analogous results hold for more refined functionals—for instance, the first conformal Laplacian eigenvalue, where optimality in the variational problem pinpoints the round sphere among all conformal classes (Petrides, 2013).

In the discrete Bakry–Émery curvature setting for weighted graphs, sharpness in the Lichnerowicz-type estimate under CD(K,∞) curvature (λ₁ ≥ K) leads to rigidity: equality is only achieved by the hypercube (or trivial weight scalings thereof) (Liu et al., 2017). The rigidity mechanism proceeds through analytic (Γ₂-calculus), geometric (distance function extension), and purely combinatorial (shell structure, small-sphere, and non-clustering) arguments, paralleling the continuum case.

A closely related discrete paradigm is the spectral rigidity of graphs in Euclidean space. For planar rigidity, algebraic connectivity conditions (μ(G) bounds) and, more recently, spectral radius (ρ(G)) thresholds provide sufficient and, in some sense, optimal criteria for graph rigidity and global rigidity. Fan–Huang–Lin establish that, for prescribed minimum degree, surpassing the spectral radius of a certain extremal graph (constructed from block-complete components and independent edge connections) guarantees rigidity, with equality classifying the unique extremal graphs (Fan et al., 2022).

2. Random Matrix Theory and Probabilistic Rigidity

In random matrix theory (RMT), eigenvalue rigidity captures the phenomenon that individual eigenvalues concentrate sharply around "classical" locations predicted by the limiting spectral measure. For Wigner, GUE, or GOE ensembles, this means that, with overwhelming probability,

$$|\lambda_i - \gamma_i| \leq C \frac{(\log N)^a}{N (\min\{i, N-i+1\})^{1/3}N^{2/3}}$$

for all $i$ not at the extreme edges, where $\gamma_i$ are quantiles of the semicircle law. Parallel results now exist for sparse or structured ensembles, such as random regular graphs (rigidity at $N^{-2/3}$ at the edge with matching GOE scaling (Huang et al., 2024)), the Jacobi Unitary Ensemble (global rigidity at an explicit scale $(\log N)/N$ normalized by the density (Dai et al., 24 Nov 2025)), and for the spectrum of random matrix minor distributions (e.g., truncations of Haar-unitaries or the LUE) (Meckes et al., 2019, Basu et al., 2019).

Technically, these results rely on determinantal point process representations, strong concentration of the eigenvalue counting function (linear statistics), and advanced probabilistic and analytic machinery (local laws, Green's function analysis, Riemann–Hilbert analysis). In each universality class, the rigidity quantitatively distinguishes the scale of eigenvalue fluctuations from their typical spacing, and is crucial for understanding extremal events, large deviation asymptotics, and correlations between eigenvalues. For instance, in the LUE, rigidity directly enables sharp large deviation estimates for the largest eigenvalue, which are essential for the analysis of transversal fluctuations in associated last-passage percolation problems (Basu et al., 2019).

3. Geometric Analysis: Rigidity in Boundary Value Problems

Eigenvalue rigidity permeates geometric analysis, particularly for PDEs under geometric constraints. For the first (or second) eigenvalue of Laplacians, Dirac operators, $p$-Laplacians, or Jacobi operators, optimal (or near-optimal) attainment of eigenvalue estimates in terms of curvature, domain shape, or conformal structure often forces the global geometry to be that of the canonical model.

Examples include:

• Rigidity for the first nonzero eigenvalue of the Laplacian on Kähler manifolds: maximal multiplicity (and spectral gap pinching) forces complex projective space (Chu et al., 2024).
• For drifting Laplacians on shrinking Ricci solitons, optimal eigenvalue pinching enforces isometry with the Gaussian soliton, or, under slightly weaker pinching, splitting into $\mathbb{R}^k$ factors (Li et al., 2024).
• For mixed boundary Laplacian problems, as soon as the first eigenvalue attains the spherical (or disk) value under Ricci or boundary curvature lower bounds, the domain is a spherical cap (Almaraz et al., 2017).
• For the first $p$-Laplacian eigenvalue among convex domains with holes and prescribed measure and perimeter, the annulus is uniquely extremal (Paoli et al., 2019).
• Quantitative spectral inequalities in domains with radial curvature bounds: equality in Steklov, Wentzell, or drifting Steklov eigenvalue bounds identifies Euclidean balls or spherically symmetric model spaces (Deng et al., 2020).

In each setting, rigidity is established through a combination of Reilly-type identities, sharp variational inequalities, symmetry or monotonicity properties, and uniqueness or classification of critical points.

4. Rigidity via Eigenvalues in Submanifold and Minimal Surface Theory

In submanifold geometry, eigenvalue rigidity is leveraged through pinching phenomena for operators encoding stability or curvature. For instance, the first or second eigenvalue of the Jacobi operator on surfaces or hypersurfaces in spheres or products attains its bound only for totally geodesic or specifically classified minimal embeddings (e.g., Clifford tori, equilateral tori, totally geodesic tori in $S^1 \times S^2$ (Batista et al., 28 May 2025), or Calabi tori in $S^{2n+1}$ via the spectrum of the fundamental matrix (Wu et al., 2024)). For constant mean curvature hypersurfaces, area or boundary-length rigidity follows from attaining the optimal upper bound on the first Jacobi eigenvalue, with equality forcing ambient and submanifold geometry to match canonical models (sphere or disk) (Batista et al., 2024).

Steklov and conformal Steklov eigenvalue rigidity for surfaces with boundary is similarly rigidifying: for annuli or Möbius bands, the supremum of the normalized first Steklov eigenvalue always exceeds the disk value $2\pi$, with the unique extremal achieved only in the flat metric, and no conformal deformation can attain the minimum (Matthiesen et al., 2020).

5. Core Techniques and Mechanisms Underlying Eigenvalue Rigidity

The pursuit and application of eigenvalue rigidity in geometry and combinatorics rely on several advanced methodologies:

• Bochner identities and curvature-dimension inequalities: Provide analytic tools for linking spectrum to curvature, both in smooth and discrete (graph) settings (Petrides, 2013, Liu et al., 2017).
• Variational methods and Rayleigh quotients: Used to identify extremal structures achieving equality and, conversely, to show the failure of non-extremal configurations to saturate spectral inequalities (Paoli et al., 2019).
• Classification of equality cases via symmetry and uniqueness results: In both manifolds and graphs, the uniqueness or "no-gap" property of extremal metrics or configurations is essential. For instance, conformal sphere, projective space, hypercube, or annulus uniqueness (Fan et al., 2022, Dai et al., 24 Nov 2025, Petrides, 2013, Paoli et al., 2019).
• Determinantal point process and RMT techniques: Vital for explicit fluctuation bounds in random matrix models, local laws, and universality classes (Meckes et al., 2019, Huang et al., 2024, Dai et al., 24 Nov 2025).
• Compactness and almost-rigidity arguments: Used when strict equality is relaxed but near-optimality still forces a Gromov–Hausdorff or metric–measure approximation to the model structure (Chu et al., 2024, Li et al., 2024).

6. Broader Implications, Universality, and Open Problems

Eigenvalue rigidity results provide a fundamental classification device across geometry, combinatorics, and probability. Their impact is seen in the uniqueness and stability of geometric models, the characterizations of extremal graphs, the universality of spectral statistics across random matrix ensembles, and the precise control of probabilistic or geometric fluctuations. As techniques advance, extensions to less regular curvature bounds, more general graphs, higher-order eigenvalues, and more intricate boundary or coupling scenarios remain active areas of research (Huang et al., 2024, Dai et al., 24 Nov 2025).

In random matrix theory, conjectures on the universality of Tracy–Widom fluctuations at the edge for sparse graphs or new ensembles, and the connection between local eigenvalue rigidity and global geometric, combinatorial, or probabilistic phenomena, remain at the forefront (Huang et al., 2024, Dai et al., 24 Nov 2025, Meckes et al., 2019). In geometric analysis and submanifold theory, similar questions about the uniqueness or multiplicity of extremal metrics or immersions (particularly for higher eigenvalues or higher codimensional settings) persist.

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Selected References:

• "Rigidity properties of the hypercube via Bakry–Émery curvature" (Liu et al., 2017)
• "Eigenvalue rigidity for truncations of random unitary matrices" (Meckes et al., 2019)
• "On a rigidity result for the first conformal eigenvalue of the Laplacian" (Petrides, 2013)
• "Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation" (Basu et al., 2019)
• "Optimal Eigenvalue Rigidity of Random Regular Graphs" (Huang et al., 2024)
• "The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound" (Chu et al., 2024)
• "Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator" (Batista et al., 28 May 2025)
• "On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble" (Dai et al., 24 Nov 2025)
• "Sharp estimates for the first $p$-Laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes" (Paoli et al., 2019)
• "A remark on the rigidity of the first conformal Steklov eigenvalue" (Matthiesen et al., 2020).