Eigenvalue Problems in Riemannian Geometry

Eigenvalue problems in Riemannian geometry lie at the intersection of analysis, topology and global geometry. At their core is the study of spectral data—eigenvalues and eigenfunctions—associated with natural differential operators such as the Laplace–Beltrami operator or its generalisations. These spectral quantities encode geometric features of manifolds, including volume growth, curvature and symmetry. Key themes include the behaviour of eigenvalue sequences under geometric constraints, sharp bounds reflecting curvature or topology, and the influence of boundary conditions on the spectrum. Beyond pure mathematics, these investigations underpin models in physics, engineering and data analysis, where the spectrum governs heat diffusion, wave propagation and quantum states. Recent advances have broadened the scope to include non-standard operators, variable boundary conditions and homogenisation limits, revealing deep connections between local geometric control and global spectral invariants. Practical applications range from shape optimisation in material science to inverse problems that seek to reconstruct geometric information from spectral measurements.

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Eigenvalue Problems in Riemannian Geometry publication trend

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Technical terms

Riemannian manifold: A smooth manifold equipped with a positive-definite metric tensor, allowing measurement of angles and distances.

Eigenvalue problem: A differential equation of the form Lf = λf, where λ is a scalar (eigenvalue) and f is a non-zero function (eigenfunction).

Laplace–Beltrami operator: The generalisation of the Laplacian to Riemannian manifolds, defined via the divergence of the gradient with respect to the metric.

Spectral gap: The difference between the first and second eigenvalues of an operator, often indicating stability and convergence rates.

Steklov problem: A boundary spectral problem in which the normal derivative of an eigenfunction on the boundary is proportional to its boundary value, with the proportionality constant as the eigenvalue.

Dirichlet boundary conditions: Constraints requiring eigenfunctions to vanish on the boundary of a domain.

References

1. Pólya’s conjecture for Euclidean balls. Inventiones Mathematicae (2023).
2. Some recent developments on the Steklov eigenvalue problem. Revista Matemática Complutense (2023).
3. Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems. Geometric and Functional Analysis (2021).