Extremal surfaces for the first Dirichlet eigenvalue

📅 martes 24 de febrero — 11:00

📍 Aula A13, Facultad de Ciencias

👤Conferenciante: Eduardo Rosinato Longa (Universidade de São Paulo)

📖 Título: Extremal surfaces for the first Dirichlet eigenvalue

📄 Resumen: While many works have investigated extremal domains for the first Dirichlet eigenvalue of the Laplacian, in this talk I will examine the corresponding problem for hypersurfaces, where one varies the immersion rather than the domain inside a fixed ambient manifold. I will present the Euler–Lagrange equation for this variational problem, which links the first eigenfunction to the second fundamental form and leads to both an interior condition and an overdetermined boundary condition. In the minimal case, these equations become particularly rigid and impose strong geometric restrictions. As an application, we show that among compact capillary CMC surfaces in the unit ball, the only extremal examples are totally geodesic discs. The proof combines variational analysis and a Hopf differential argument.