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Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

Tirumala Chakradhar

TL;DR

This work extends the Steklov problem to differential $p$-forms on warped product manifolds, focusing on the Dirichlet-to-Neumann spectrum of $\Lambda^{(p)}$ and employing separation of variables to obtain sharp geometric bounds. Under nonnegative Ricci curvature and strictly convex boundary, it proves Escobar-type lower bounds $\bm{\sigma}^{(p)}_{(m)}(M) \ge (m+p)\kappa$ and identifies rigidity in equality cases, including Euclidean-ball geometry; for hypersurfaces of revolution, it reveals isospectrality in even dimensions when $p=(n-2)/2$ and derives a range of bounds depending on meridian length when there are two boundary components. The analysis yields ratio and gap bounds $\bm{\sigma}^{(p)}_{(k+1)}/\bm{\sigma}^{(p)}_{(k)} \le \lambda^{(p)}_{(k+1)}/\lambda^{(p)}_{(k)}$ and connects cylinder and ball spectra to the warped-product geometry, extending function-case results to general $p$-forms and highlighting the role of topology and warping in spectral bounds.

Abstract

We consider the Steklov problem on differential $p$-forms defined by M. Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $0$-forms), highlighting the influence of the underlying topology on the spectrum for $p$-forms in general.

Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

TL;DR

This work extends the Steklov problem to differential -forms on warped product manifolds, focusing on the Dirichlet-to-Neumann spectrum of and employing separation of variables to obtain sharp geometric bounds. Under nonnegative Ricci curvature and strictly convex boundary, it proves Escobar-type lower bounds and identifies rigidity in equality cases, including Euclidean-ball geometry; for hypersurfaces of revolution, it reveals isospectrality in even dimensions when and derives a range of bounds depending on meridian length when there are two boundary components. The analysis yields ratio and gap bounds and connects cylinder and ball spectra to the warped-product geometry, extending function-case results to general -forms and highlighting the role of topology and warping in spectral bounds.

Abstract

We consider the Steklov problem on differential -forms defined by M. Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., -forms), highlighting the influence of the underlying topology on the spectrum for -forms in general.

Paper Structure

This paper contains 8 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Steklov problem on differential forms
  5. Warped product manifolds
  6. Proof of Theorem 1.2
  7. Proofs of Theorems 1.4 and 1.5
  8. Proofs of Theorems 1.6, 1.7 and 1.8

Key Result

Theorem 1.2

Let $(M^n,g)$ be a warped product manifold as in Section subsec: warped prod mfds ($n\geq 2$), such that $\mathop{\mathrm{Ric}}\nolimits_g\geq 0$ and it has a strictly convex boundary. If $\kappa\in \mathbb{R}_{>0}$ is the principal curvature of ${\partial M}$, then, for $p\leq \frac{n-1}{2}$, the $ with the equality holding for a given $m\in \mathbb{N}$ if and only if $M$ is the Euclidean ball of

Theorems & Definitions (17)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 2.1
  • ...and 7 more