Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds

2025/12/28 by Lv, Zongyi, Xiong, Changwei, Zou, Yuxun
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2512.22885

Abstract

In the first part, we derive monotonicity of the normalized spectra for the second-order Steklov problem and two fourth-order Steklov problems on the 2-dimensional geodesic disks with respect to the geodesic radius in the sphere and the hyperbolic space. The normalizations are made using four natural geometric factors. As corollaries, we get Escobar-type bounds for Steklov eigenvalues on 2-dimensional geodesic disks with varying curvature in space forms. We also get two monotonicity results for higher-dimensional cases. In the second part, we obtain some sharp bounds concerning the spectra of the two fourth-order Steklov problems on warped product manifolds with non-negative Ricci curvature and a strictly convex boundary. In particular, we confirm Qiaoling Wang and Changyu Xia's conjecture (2018) on the sharp lower bound of the first non-zero eigenvalue of a fourth-order Steklov problem in the case of 3-dimensional warped product manifolds.

Citations

• Scaling inequalities and limits for Robin and Dirichlet eigenvalues
• A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates
• Some recent developments on the Steklov eigenvalue problem
• The Bilaplacian with Robin boundary conditions
• Escobar's Conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue
• On the spectra of three Steklov eigenvalue problems on warped product manifolds
• Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator
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• Sharp bounds for the first eigenvalue of a fourth order Steklov problem
• Inequalities for the Steklov Eigenvalues
• A note on the first eigenvalue of spherically symmetric manifolds
• Inequalities for membrane and Stekloff eigenvalues

Related

• Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds
• On the spectra of three Steklov eigenvalue problems on warped product manifolds
• Upper bounds for the Steklov eigenvalues of warped products
• Upper bound for Steklov eigenvalues of warped products with fiber of dimension 2
• Sharp bounds for the first eigenvalue of a fourth order Steklov problem