arXiv subjects

Jade Brisson

Publications and source records attributed to Jade Brisson.

6 recordsLinked to original sources

arXiv2025

Upper bounds for the Steklov eigenvalues of warped products

We obtain upper bounds for the Steklov eigenvalues of warped products $\Omega\times_h\Sigma$, where $\Omega$ is a compact Riemannian manifold with boundary and $\Sigma$ is a closed Riemannian manifold. These bounds involve the volume of $\Omega$ and of $\partial\Omega$ as well as the eigenvalues of the Laplace operator on the fiber $\Sigma$ and the $L^p$-norm of the warping function $h$. The bounds are very different depending on the dimension $n$ of the fiber $\Sigma$ and the value of $p$. In some cases, we obtain optimal upper bounds and stability estimates.

math.SP↗

arXiv2024

Upper bound for Steklov eigenvalues of warped products with fiber of dimension 2

In this note, we investigate the Steklov spectrum of the warped product $[0,L]\times_h \Sigma$ equipped with the metric $dt^2+h(t)^2g_\Sigma$, where $\Sigma$ is a compact surface. We find sharp upper bounds for the Steklov eigenvalues in terms of the eigenvalues of the Laplacian on $\Sigma$. We apply our method to the case of metric of revolution on the 3-dimensional ball and we obtain a sharp estimate on the spectral gap between two consecutive Steklov eigenvalues.

math.SP↗

arXiv2024

Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics

We investigate upper bounds for the spectral ratios and gaps for the Steklov eigenvalues of balls with revolution-type metrics. We do not impose conditions on the Ricci curvature or on the convexity of the boundary. We obtain optimal upper bounds for the Steklov spectral ratios in dimensions 3 and higher. In dimension 3, we also obtain optimal upper bounds for the Steklov spectral gaps. By imposing additional constraints on the metric, we obtain upper bounds for the Steklov spectral gaps in dimensions 4 and higher.

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arXiv2023

Tubes and Steklov eigenvalues in negatively curved manifolds

We consider the Steklov eigenvalue problem on a compact pinched negatively curved manifold $M$ of dimension at least three with totally geodesic boundaries. We obtain a geometric lower bound for the first nonzero Steklov eigenvalue in terms of the total volume of $M$ and the volume of its boundary. We provide examples illustrating the necessity of these geometric quantities in the lower bound. Our result can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalue on closed pinched negatively curved manifolds of dimension at least three proved by Schoen in 1982. The proof is composed of certain key elements. We provide a uniform lower bound for the first eigenvalue of the Steklov-Dirichlet problem on a neighborhood of the boundary of $M$ and show that it provides an obstruction to having a small first nonzero Steklov eigenvalue. As another key element of the proof, we give a tubular neighborhood theorem for totally geodesic hypersurfaces in a pinched negatively curved manifold. We give an explicit dependence for the width function in terms of the volume of the boundary and the pinching constant.

math.DG↗

arXiv2023

Multiple tubular excisions and large Steklov eigenvalues

Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the first non-zero eigenvalue of the domain $\Omega_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around each $N_j$ tends to infinity as $\varepsilon$ tends to $0$. More precisely, we prove a lower bound in terms of $\varepsilon$, $b$, the geometry of $M$ and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\varepsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b\,,b+1\,,\ldots$, we have a stronger result: their order of divergence is $\varepsilon^{-1}$ and their rate of divergence is only depending on $m$ and on the codimensions of the submanifolds.

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arXiv2021

Tubular excision and Steklov eigenvalues

Given a closed manifold $M$ and a closed connected submanifold $N\subset M$ of positive codimension, we study the Steklov spectrum of the domain $\Omega_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around $N$. All non-zero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of $N$ in $M$. Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.

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