Date: Friday, September 15, 2017
Location: 3866 East Hall (3:00 PM to 5:00 PM)
Title: Large gaps for Steklov eigenvalues under fixed boundary geometry
Abstract: This talk will focus on the spectral geometry of Steklov eigenvalues. Steklov eigenvalues are eigenvalues of the DirichlettoNeumann operator, which maps a function on the boundary of a Riemannian manifold to the outward normal derivative of the harmonic extension of that function. Such operators are essential in electrical impedance tomography (the DirichlettoNeumann operator is also known as the voltagetocurrent map). Since they are eigenvalues of an operator defined on the boundary of a manifold, naturally one might expect that the Steklov eigenvalues of a manifold are closely related to the geometry of the boundary. Indeed, the Steklov spectral asymptotics are completely determined by the geometry in a neighborhood of the boundary. Despite this, Colbois, El Soufi, and Girouard [arXiv:1701.04125] recently constructed conformal deformations of a given metric under which the boundary geometry remains fixed, but all nontrivial Steklov eigenvalues tend to infinity. In this talk I will present two constructions inspired by the conformal deformations found in [arXiv:1701.04125]. The first is a conformal deformation of a given metric that is localized in an arbitrary neighborhood of the boundary that keeps the boundary geometry fixed. I will show that the boundedness of the Steklov spectrum with regards to these deformations depends on the number of connected components of the domain on which we localize. Secondly, all of the conformal deformations previously mentioned have the peculiar property that the volume inside the manifold tends to infinity (this is a necessary condition for conformal deformations whose Steklov eigenvalues are unbounded). I will present a family of metrics, no longer in the same conformal class, that send the entire Steklov spectrum to infinity, but keep the boundary geometry and volume inside the manifold fixed. This work is joint with Alexandre Girouard.
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Speaker: Donato Cianci
Institution: U Michigan
Event Organizer: Spatzier
