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.

The Mean Curvature Flow (MCF) describes the evolution of hypersurfaces in Euclidean space, driven by their mean curvature, which tends to smooth out geometric irregularities over time. However, singularities inevitably develop during the flow, marking critical points where the smooth evolution ceases. In this talk, we will examine the formation of singularities in MCF, focusing on the crucial role of tangent flows in their analysis. Tangent flows, which emerge as blow-up limits near singularities, often exhibit self-similar structures. We will highlight how the mean curvature flow produces a specific type of tangent flow at the first singularity, preserving notable geometric and topological properties of the compact initial data. This presentation is based on an ongoing work with David Hoffman and Brian White.

The k-Yamabe problem is a fully non-linear extension of the classical Yamabe problem that seeks for metrics of constant k-curvature. In this talk I will discuss this equation from the point of view of geometric flows and provide existence and classification results on soliton solutions for the k-Yamabe flow. This is joint work with Maria Fernanda Espinal

We discuss an eigenvalue estimate that holds on every embedded self-similar shrinker for mean curvature flow. This result is obtained via a Reilly-type formula and can be viewed as an analogue of the first eigenvalue estimate obtained by Choi and Wang for embedded minimal hypersurfaces in the round sphere. Our estimate generalizes earlier work of Ding and Xin on closed self-shrinkers by introducing and minimizing a new functional to treat the non-compact case. This is joint work with Simon Brendle.