### Tubes in negatively curved manifold and Steklov eigenvalues

Analysis and Geometry Seminar

19th October 2023, 3:30 pm – 4:30 pm

Fry Building, Room 2.04

Any closed simple geodesic on a hyperbolic surface has an embedded collar neighbourhood whose width only depends on the length of the geodesic. This is the well-known collar lemma. In this talk I will present a generalisation of this results of this result to higher dimension and variable negative curvature manifolds: any closed totally geodesic hypersurface on a manifold with pinched negative curvature has a tubular neighbourhood whose width depends only on the volume of the hypersurface and bounds on the curvature of the manifold. Finally, I will briefly discuss how those results can be applied to obtain bounds for Steklov eigenvalues problem. This is ongoing work with A. Basmajian, J. Brisson and A. Hassannezhad.

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