Alexandre Girouard

The Steklov spectrum of surfaces: Asymptotics and invariants

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Isoperimetric control of the spectrum of a compact hypersurface

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.

Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces

We study the effect of two types of degeneration of a Riemannian metric on the first eigen- value of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

The spectral gap of graphs and Steklov eigenvalues on surfaces

Using expander graphs, we construct a sequence

Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

\{\Omega_N\}_{N\in\mathbb{N}}

Large spectral gaps for Steklov eigenvalues under volume constraints and under localized conformal deformations

of smooth compact surfaces with boundary of perimeter

SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM

N

Isoperimetric control of the Steklov spectrum

, and with the first non-zero Steklov eigenvalue

Maximization of the second positive Neumann eigenvalue for planar domains

\sigma_1(\Omega_N)