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The study of eigenfunctions of the Laplacian on Riemannian manifolds is a classical topic in differential geometry, closely related to the Uniformization Theorem. This theorem classifies simply connected Riemannian surfaces into three conformal types: elliptic (sphere), parabolic (plane), and hyperbolic (unit disk). Since harmonic functions are conformally invariant in dimension two, this classification determines whether a non-compact surface admits a rich set of bounded harmonic functions or only trivial ones, as in the Euclidean plane. Motivated by extending this idea to higher-dimensional manifolds, recent research has focused on the study of harmonic functions and solutions to the eigenvalue problem $\Delta u + \lambda u = 0$ (called $\lambda$-harmonic functions). In this talk, we introduce the theory of bounded harmonic functions on Hadamard manifolds (simply connected with negative curvature) and explore the existence of $\lambda$-harmonic functions with zero Dirichlet data on a domain $\Omega \subset M$. As we will see, this existence is strongly linked to the presence of certain convex sets with specific geometric properties. This talk is based on an ongoing work with José Espinar and Marcos Petrucio.

In this talk we present a result about a classification of the rotationally-symmetric solutions to an overdetermined problem in the 2-sphere. As a consequence, we give a rigidity result about a certain type of minimal surfaces and as a corollary of this result we provide a new characterization of the critical catenoid among the embedded free boundary minimal annulus in the unit ball. This is based in a join work with José M. Espinar.