[Paper Review] The first Stekloff eigenvalue in weighted Riemannian manifolds

Márcio Batista, J. I. Santos|arXiv (Cornell University)|Apr 10, 2015

Nonlinear Partial Differential Equations12 references3 citations

TL;DR

This paper establishes sharp estimates for the first non-zero Stekloff eigenvalue in weighted Riemannian manifolds using Bakry-Émery Ricci curvature bounds and weighted mean curvature. It proves that under non-negative weighted Ricci curvature and lower bounds on weighted mean curvature, the first Stekloff eigenvalue is bounded above or below by geometric quantities, with equality only for Euclidean balls with constant weight functions, extending Escobar's classical results to the weighted setting.

ABSTRACT

In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper we are interested to study this problem in weighted context. Various estimates are obtained, including a sharp estimate on surfaces, similar to that Escobar obtained.

Motivation & Objective

• To extend Escobar's sharp eigenvalue estimate for the Stekloff problem on surfaces to the weighted Riemannian setting.
• To analyze the first non-zero Stekloff eigenvalue in compact weighted manifolds with boundary under curvature and mean curvature constraints.
• To derive upper and lower bounds for the first Stekloff eigenvalue in terms of geometric and weighted curvature invariants.
• To characterize equality cases, showing that equality holds only for Euclidean balls with constant weight functions.

Proposed method

• Utilizes the weighted Laplacian (Drift Laplacian) Δ_f = Δu − ⟨∇u, ∇f⟩ on compact Riemannian manifolds with boundary.
• Applies Bakry-Émery Ricci curvature bounds Ric_f^k ≥ 0 and weighted mean curvature H_f ≥ c to derive eigenvalue estimates.
• Employs variational methods and maximum principle arguments on eigenfunctions to derive inequalities involving the first Stekloff eigenvalue p_1 or q_1.
• Uses the second fundamental form II and geodesic curvature k_g on the boundary to relate intrinsic geometry to eigenvalue bounds.
• Imposes boundary conditions Δ_f u = 0 and ∂u/∂ν = p_1 u (or higher-order variants) to define the Stekloff eigenvalue problem.
• Analyzes equality cases via curvature rigidity, showing that equality implies isometry to a Euclidean ball with constant f and k = n+1.

Experimental results

Research questions

• RQ1 What are sharp upper and lower bounds for the first non-zero Stekloff eigenvalue in weighted Riemannian manifolds with Ric_f^k ≥ 0 and H_f ≥ c?
• RQ2 Under what geometric and curvature conditions does the first Stekloff eigenvalue achieve equality in the derived bounds?
• RQ3 How does the inclusion of a smooth weight function f modify the classical Stekloff eigenvalue estimates of Escobar for surfaces?
• RQ4 Can the equality case in the eigenvalue estimates be characterized, and what does it imply about the manifold’s geometry?
• RQ5 What is the role of the weighted mean curvature H_f = H − (1/n)⟨ν, ∇f⟩ in determining the eigenvalue bounds?

Key findings

• The first Stekloff eigenvalue p_1 satisfies p_1 ≤ √λ_1 / ((k−1)c) (√λ_1 + √(λ_1 − (k−1)c²)) under Ric_f^k ≥ 0, H_f ≥ (k−1)c/n, and II ≥ cI, with equality iff M is an n-dimensional Euclidean ball of radius 1/c, f is constant, and k = n+1.
• For the second-order Stekloff problem, q_1 ≥ nc holds under Ric_f^k ≥ 0 and H_f ≥ ((k−1)/k)c, with equality iff M is an (n+1)-dimensional Euclidean ball, f is constant, and k = n+1.
• The first eigenvalue q_1 is bounded above by A/V, the ratio of weighted area to weighted volume, with equality implying M is isometric to a Euclidean ball, f constant, and k = n+1.
• When ν coincides with the normalized gradient of f on ∂M, the eigenvalue satisfies p_1 > k_g − f_ν ≥ c, and equality occurs when k_g = k_0 and f_ν = 0.
• In the equality case, the Gaussian curvature vanishes, and the manifold is isometric to a Euclidean ball with constant f, implying rigidity of the geometry under curvature and boundary conditions.
• The proof relies on the vanishing of Hess φ and Ric_f(∇φ, ∇φ) when v is constant, leading to the conclusion that the manifold is flat and the boundary is totally umbilic with constant geodesic curvature.