Conformal Geometry and Spectral Bounds on Manifolds with Boundary

TL;DR

The paper develops Korevaar-type, conformal-volume-based upper bounds for higher Steklov eigenvalues on manifolds with boundary, expressed in terms of the relative conformal volume $V_{rc}$ and the isoperimetric ratio $I(M)$. It extends these bounds to the conformal Dirichlet-to-Robin spectrum on balls and, more generally, to manifolds with proper conformal immersions into Euclidean balls, while proving a Hersch-type extremal result showing the Euclidean ball metric maximizes the first eigenvalue in its conformal class. A unified variational framework and a construction of test functions supported on many disjoint annuli underpin the estimates, with key applications to the counting of negative eigenvalues and to the Morse index of type-II stationary capillary hypersurfaces. The results connect spectral bounds with geometric invariants and yield concrete lower bounds for negative eigenvalues in terms of $V_{rc}$, volume, and curvature data, providing tools for capillary geometry and Yamabe-type problems on manifolds with boundary.

Abstract

This work investigates upper bounds for the spectrum of the Steklov-type operator on Riemannian manifolds with boundary. We extend the Fraser-Schoen estimate for the first positive Steklov eigenvalue to higher Steklov eigenvalues, in terms of the relative conformal volume and the isoperimetric ratio. Our approach, which draw on Korevaar's method, further developed by Grigor'yan-Netrusov-Yau and Kokarev, can be adapt to derive a Korevaar-type estimate for the conformal Dirichlet-to-Robin operator on the Euclidean ball, showing that its $k$-th eigenvalue is bounded from above by a multiple of $k^{2/n}$, as well as a corresponding bound in terms of the relative conformal volume for proper conformal immersion into the Euclidean ball. We also establish a lower bound for the number of negative eigenvalues of the Steklov-type problem in terms of the relative conformal volume, with applications to the spectrum of the conformal Dirichlet-to-Robin operator and to the Morse index of type-II stationary capillary hypersurfaces.

Theorems & Definitions (34)

• Remark 2.1
• Proposition 3.1: Grigor’yan–Netrusov–Yau gny
• Lemma 3.2
• proof
• Definition 4.1
• Theorem 4.2
• Corollary 4.3
• Remark 4.4
• Remark 4.5
• proof : Proof of Theorem \ref{['thm:letterA']}