Steklov Eigenvalues on Warped Products

• The paper introduces a framework utilizing separation of variables to express Steklov eigenvalues on warped products as solutions to auxiliary weighted problems.
• Sharp quantitative upper and lower eigenvalue bounds are derived using the interplay between the warping function, base geometry, and fiber spectral properties.
• Stability estimates and inverse spectral results demonstrate how geometric data are recovered, offering new perspectives on boundary value problems in spectral geometry.

A warped product manifold provides a fundamental construction in global analysis and geometry, producing new Riemannian manifolds from simpler pieces: a base, a fiber, and a warping function. The Steklov problem, an eigenvalue boundary value problem linked to the Dirichlet-to-Neumann map, exhibits rich structure on warped products, particularly due to the interplay between the geometry of the base, the spectral properties of the fiber, and the warping function controlling the local scaling. Recent advances provide sharp quantitative upper and lower bounds, stability estimates, and precise asymptotics for Steklov eigenvalues on these manifolds, revealing deep connections with spectral geometry, convexity, and volume concentration.

1. Warped Product Geometry and the Steklov Problem

Let $(\Omega, g_\Omega)$ be a compact Riemannian manifold with boundary $\partial\Omega$, and %%%%2%%%% a closed Riemannian manifold. For a smooth positive warping function $h:\Omega \to (0, \infty)$, the warped product is

$$M_h = \Omega \times_h \Sigma, \qquad g_h(x,y) = g_\Omega(x) + h(x)^2\,g_\Sigma(y).$$

The Steklov problem on $(M_h, g_h)$ seeks nontrivial $u \in C^\infty(M_h)$ satisfying

$$\Delta_{g_h}u = 0 \quad \text{in } M_h, \qquad \partial_\nu u = \sigma u \quad \text{on } \partial M_h = \partial\Omega \times \Sigma,$$

where $\partial_\nu$ denotes the outward normal derivative. The spectrum $$0 = \sigma_0 < \sigma_1 \leq \sigma_2 \leq \cdots \to \infty$$ is discrete.

A fundamental property is that by separation of variables, the Steklov spectrum of $M_h$ is explicitly computable in terms of auxiliary weighted problems on the base $\Omega$ for each Laplace eigenvalue $\lambda_j$ on $\Sigma$: $$-\Delta a - \frac{n}{h}\langle \nabla h, \nabla a\rangle + \lambda_j h^{-2} a = 0 \text{ in } \Omega, \qquad \partial_\nu a = \sigma a \text{ on } \partial\Omega,$$ with Rayleigh quotient

$$R_{\lambda, h}(a) = \frac{\int_\Omega (|\nabla a|^2 h^n + \lambda a^2 h^{n-2})\,dV_\Omega}{\int_{\partial\Omega} a^2 h^n\,dA_\Omega}.$$

The Steklov spectrum of $M_h$ is the union of the solutions for all $\lambda_j$ (Brisson et al., 17 Dec 2025).

2. Universal and Sharp Bounds for Steklov Eigenvalues

The structure of the Rayleigh quotient gives rise to general upper bounds depending on the dimension $n = \dim \Sigma$, the Laplace eigenvalues $\lambda_j$, $L^p$-norms of $h$, and geometric measures of $(\Omega, \partial\Omega)$. For $n \geq 2$ (Brisson et al., 17 Dec 2025):

• For any $k \ge 1$ and $h > 0$,

$$\sigma_k(M_h) < \frac{\lambda_k \int_\Omega h^{n-2} dV_\Omega}{\int_{\partial\Omega} h^n dA_\Omega}$$

(strict inequality, equality only if $\lambda_k = 0$).

• For $h \leq C$ on $\Omega$ and $h \equiv 1$ on $\partial\Omega$,

$$\sigma_k(M_h) < C^n \sigma_k(M_C) < C^{n-2} \frac{\lambda_k |\Omega|}{|\partial\Omega|}.$$

• For $n=2$ and $h \leq C$, $h|_{\partial\Omega} = 1$,

$$\sigma_k(M_h) < C^2 \sigma_k(M_C) < \frac{\lambda_k |\Omega|}{|\partial\Omega|}.$$

The sharp supremal value over all $h$ with $h|_{\partial\Omega}=1$ is $\frac{\lambda_k |\Omega|}{|\partial\Omega|}$, but this is only approached in a blowup regime.

For higher dimensions ($n \ge 3$), the eigenvalues' growth with respect to the $L^p$ norm of $h$ exhibits phase transitions:

• For $p > n-2$,

$$\sigma_k(M_h) < \frac{\lambda_k |\Omega|^{(p-(n-2))/p}}{|\partial\Omega|} \|h\|_{L^p(\Omega)}^{n-2}.$$

• For $p = n-2$, the bound involves only $\|h\|_{L^{n-2}}$, while for $p < n-2$ (and connected $\partial\Omega$), it is possible to construct families with $\|h\|_{L^p}$ bounded but $\sigma_k(M_h) \to \infty$, indicating that no uniform bound can be achieved for such $p$ (Brisson et al., 17 Dec 2025).

In the codimension-one case with a one-dimensional base ($\Omega = [0, L]$), bounds are uniform for all $p \ge 1$: $$\sigma_k(M_h) \le \text{const} \cdot \|h\|_{L^p}^{n-2} L^{(p - (n-2))/p} \lambda_k.$$

3. Saturation, Stability, and Volume Concentration

Sharpness of the upper bounds is attained in an asymptotic (singular) regime. For fixed $C \ge 1$ and $h = 1$ near $\partial\Omega$, $h = C$ on most of $\Omega$, and an interpolation layer of vanishing thickness, one has

$$\sup \{\sigma_k(M_h): h \le C, h = 1 \text{ on } \partial\Omega\} = C^n \sigma_k(M_C),$$

and as $C \to \infty$, $$C^2 \sigma_k(M_C) \to \lambda_k \frac{|\Omega|}{|\partial\Omega|}$$ (Brisson et al., 17 Dec 2025).

For surfaces ($n=2$), the upper bound can only be approached by making $\int_D h^2 \to \infty$ on every interior subdomain $D \subset \Omega$, leading to so-called blowup-stability phenomena. Quantitative estimates relate how close $\sigma_k(M_h)$ is to the optimal bound and how much $h^2$ must concentrate internally.

Stability results show, for example, that if $$\delta(h) = \lambda_k \frac{|\Omega|}{|\partial\Omega|} - \sigma_k(M_h)$$ is small, then any set $D$ of positive measure must have $\int_D h^2$ large, with explicit lower bounds in terms of $\delta(h)$ (Brisson et al., 17 Dec 2025).

4. Separation of Variables and Spectral Reduction

The computation of Steklov spectra on warped products proceeds via separation of variables:

• The Laplacian $\Delta_{g_h}$ operator in these products decomposes, yielding a family of coupled PDE-ODE problems.
• Fixing Laplace eigenfunctions $\phi_j$ on $\Sigma$, harmonic functions on $M_h$ are written as $u(x, y) = a_j(x) \phi_j(y)$.
• For functions, the resulting ODE or PDE in $a_j$ produces Sturm–Liouville-type problems or weighted Steklov–Helmholtz boundary value problems.

The structure of the Steklov spectrum is detailed:

• The spectrum of $M_h$ is the union over all base auxiliary problems indexed by the fiber Laplacian's spectrum.
• For model cases (e.g., $\Omega = [0, L]$, $\Sigma$ a surface, $h(t)$ with $h(0) = h(L) = 1$), the variational characterization yields sharp explicit upper bounds (Brisson et al., 20 Mar 2024).
• This reduction underpins both the quantitative analysis (bounds, asymptotics, stability) and spectral recovery results (inverse problems).

5. Inverse Problems and Determination of the Warping Function

For warped product balls $[0,R] \times S^{d-1}$ or similar models, the Steklov spectrum encodes the warping factor up to isometry. Precisely:

• The full Steklov spectrum determines the warping function $f$ uniquely, globally, and stably (up to mild regularity/corner assumptions).
• Approximate spectral data (subject to explicit decay rates) determine $f$ near the boundary, with log-type stability.
• The key bridge is the mapping from spectral data to the Weyl–Titchmarsh $m$-function, and then to a Laplace transform in radial variable, which via moment inversion recovers the "radial potential," then $f$ via an ODE with Cauchy data at the boundary (Daudé et al., 2018).

This connection enables precise uniqueness and stability results in the high-frequency regime, confirming that the spectral data encode comprehensive geometric information about the warped structure.

6. Extensions: Differential Forms, Higher-Order Problems, and Localization Phenomena

Recent work generalizes the analysis to:

• Steklov problems on coclosed $p$-forms, with Rayleigh quotients incorporating topology; sharp Escobar-type lower bounds are found in terms of boundary principal curvature, and topological shifts (eigenvalues bound below by $(m+p)K$) appear for $2p+1 < n$ (Chakradhar, 28 Oct 2024).
• Fourth-order Steklov-type problems, where lower (and upper) bounds depend on warping-derivative ratios (e.g., $h'(R)/h^3(R)$) and are sharp for Euclidean balls (Xiong, 2019).
• Localization of Steklov eigenfunctions in products with two boundary components. High-frequency eigenmodes may localize exponentially near distinct boundary components (the "flea on the elephant" phenomenon), particularly evident under symmetric or near-symmetric warping (Daudé et al., 2021).

These results illuminate the spectral-geometric richness of Steklov-type problems beyond the scalar case.

7. Illustrative Models and Limiting Cases

The theory is elucidated by several canonical models:

• For $\Sigma = S^n$ (round sphere) or $T^n$ (flat torus), Laplace eigenvalues' scaling governs growth rates of Steklov eigenvalues with respect to warping norms (Brisson et al., 17 Dec 2025).
• For constant warping ($h \equiv C$), the spectrum is rescaled accordingly, recovering asymptotic behavior.
• On the interval base with any $L^p$ integrable warping, all Steklov eigenvalues remain bounded (Brisson et al., 17 Dec 2025).
• For revolution metrics on the ball, optimality and sharp gap estimates are achieved (Brisson et al., 20 Mar 2024).
• The Euclidean ball is uniquely characterized by simultaneous saturation of all spectral bounds under Ricci nonnegativity and boundary convexity (Chakradhar, 28 Oct 2024, Xiong, 2019).

The synthesis of explicit constructions, variational characterizations, and sharp analysis grounds the field of Steklov eigenvalues on warped products in both theoretical and practical geometry.

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Relevant references:

• (Brisson et al., 17 Dec 2025) Upper bounds for the Steklov eigenvalues of warped products
• (Girouard et al., 13 Mar 2024) Large Steklov eigenvalues under volume constraints
• (Brisson et al., 20 Mar 2024) Upper bound for Steklov eigenvalues of warped products with fiber of dimension 2
• (Daudé et al., 2021) Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon
• (Chakradhar, 28 Oct 2024) Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds
• (Daudé et al., 2018) Stability in the inverse Steklov problem on warped product Riemannian manifolds
• (Xiong, 2019) On the spectra of three Steklov eigenvalue problems on warped product manifolds