Steklov Eigenvalues in Negatively Curved Manifolds

Explore a seminar on Spectral Geometry focusing on Steklov eigenvalues in negatively curved manifolds. Delve into the Steklov eigenvalue problem on compact pinched negatively curved manifolds with totally geodesic boundaries. Discover how the first nonzero Steklov eigenvalue is bounded below in terms of total volume and boundary area for dimensions three and higher. Learn about the implications of this finding, including how Steklov eigenvalues can only approach zero as total volume and/or boundary area approach infinity. Compare this result to Schoen's 1982 proof of lower bounds for first nonzero Laplace eigenvalues on closed pinched negatively curved manifolds. Gain insights from the collaborative research of Jade Brisson, Ara Basmajian, Asma Hassannezhad, and Antoine Métras in this 38-minute presentation from the Centre de recherches mathématiques (CRM).