Eigenvalues and constructions of minimal surfaces

Eigenvalues of the Laplace operator of Euclidean domains govern many
physical phenomena, including heat flow and sound propagation. In
particular, various inequalities for Laplace eigenvalues have
fascinated mathematicians since the XIXth century. The following
question was first formulated by Lord Rayleigh in his “Theory of
sound”: which planar domain of a given area has the lowest first
Dirichlet eigenvalue? This is an example of an isoperimetric
eigenvalue problem for planar domains. The focus of the present talk
is on more general isoperimetric problems, where one considers
surfaces equipped with Riemannian metrics. More specifically, sharp
upper bounds for Laplace and Steklov eigenvalues have been an active
area of research for the past decade, largely due to their fascinating
connection to fundamental geometric objects, minimal surfaces. We will
survey recent results exploring the applications of this connection
both to minimal surface theory and to isoperimetric eigenvalue
problems, culminating in recent powerful applications to the
construction of novel minimal surfaces in the 3-sphere and 3-ball.