[Paper Review] A general discrete Wirtinger inequality and spectra of discrete Laplacians

Ivan Izmestiev|arXiv (Cornell University)|Feb 11, 2015

Spectral Theory in Mathematical Physics13 references3 citations

TL;DR

This paper generalizes the discrete Wirtinger inequality by introducing a weighted, geometrically motivated version involving angles αᵢ on a circle, linking it to the spectral gap of a discrete Laplacian. The key contribution is a new inequality and matrix signature theorem that extend classical results and suggest a discrete analog of the Lichnerowicz-Obata theorem for spherical cone-metrics with positive curvature.

ABSTRACT

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In higher dimensions, the mixed volumes theory leads to similar results, which allows us to associate a discrete Laplace operator to every geodesic triangulation of the sphere and, by analogy, to every triangulated spherical cone-metric. For a cone-metric with positive singular curvatures, we conjecture an estimate on the spectral gap similar to the Lichnerowicz-Obata theorem.

Motivation & Objective

• To generalize the Fan-Taussky-Todd discrete Wirtinger inequality to a weighted, geometric setting involving variable angles αᵢ on a circle.
• To establish a connection between the spectral gap of a weighted discrete Laplacian and a geometric isoperimetric problem on a cone.
• To extend the framework to higher dimensions using mixed volumes and quermassintegrals, leading to a discrete analog of the Lichnerowicz-Obata theorem.
• To conjecture a discrete Lichnerowicz-type estimate for spherical cone-metrics with positive singular curvature.

Proposed method

• Derives a general discrete Wirtinger-type inequality (Theorem 2) involving weighted differences (xᵢ − xᵢ₊₁)² / sin αᵢ₊₁ and quadratic forms with tan(αᵢ/2) coefficients.
• Proves a signature theorem (Theorem 3) for a circulant tridiagonal matrix M associated with the quadratic form, showing its eigenvalue signature depends on the total angle sum ∑αᵢ modulo 2π.
• Uses a geometric construction related to the discrete isoperimetric problem on a cone, where the optimal polygon encloses maximal area under fixed side directions and perimeter.
• Applies tools from mixed volumes and quermassintegrals in higher dimensions to define a discrete Laplacian operator via the matrix M for polyhedral and triangulated spherical metrics.
• Establishes a discrete analog of the Weitzenböck formula by analyzing the negative semidefiniteness of quadratic forms on links of vertices in Delaunay triangulations.
• Conjectures that for spherical cone-metrics with all cone angles < 2π, the quadratic form ⟨Mx, x⟩ ≤ 0 whenever ⟨Mx, 1⟩ = 0, extending the Lichnerowicz-Obata theorem.

Experimental results

Research questions

• RQ1 What is the optimal constant in a generalized discrete Wirtinger inequality with variable weights αᵢ on a circular lattice?
• RQ2 How does the spectral gap of a weighted discrete Laplacian relate to geometric constraints such as total angle sum and cone curvature?
• RQ3 Can the classical Lichnerowicz-Obata theorem on the spectral gap of the Laplacian on spheres be discretized for spherical cone-metrics?
• RQ4 What conditions on a triangulated spherical metric ensure that the associated discrete Laplacian satisfies a discrete Lichnerowicz-type spectral gap estimate?

Key findings

• The generalized discrete Wirtinger inequality (Theorem 2) holds if and only if the total angle sum ∑αᵢ ≤ 2π, with equality case corresponding to harmonic functions of the form xₖ = a cos(∑ᵢ₌₁ᵏ αᵢ) + b sin(∑ᵢ₌₁ᵏ αᵢ).
• The circulant tridiagonal matrix M defined by the angles αᵢ has a precise signature (p, q, r) depending on whether ∑αᵢ = 2mπ or lies strictly between 2mπ and 2(m+1)π.
• When ∑αᵢ = 2π, the kernel of M consists exactly of vectors of the form xₖ = a cos(∑ᵢ₌₁ᵏ αᵢ) + b sin(∑ᵢ₌₁ᵏ αᵢ), corresponding to the harmonic functions on the circle.
• For ∑αᵢ > 2π, the inequality fails for certain non-zero xᵢ, indicating a sharp threshold in the geometric parameter space.
• In higher dimensions, the quermassintegral Wₙ₋₂ defines a quadratic form Wₙ₋₂(h) = ⟨Mh, h⟩, where M is a symmetric matrix derived from facet normals νᵢ and support numbers hᵢ.
• The discrete Lichnerowicz conjecture (Conjecture 5.3) posits that for Delaunay triangulations of spherical cone-metrics with all cone angles < 2π, ⟨Mx, 1⟩ = 0 implies ⟨Mx, x⟩ ≤ 0, with equality iff x is a linear functional on the normals.