Title: Some rigidity results for charged Riemannian manifolds

Abstract: In this lecture, we investigate the geometric consequences of equality in Gibbons' area–charge inequality for stable minimal 2-spheres Σ² arising in the Einstein–Maxwell setting. Under suitable energy (curvature) assumptions, we show that saturation of the inequality

A(Σ) ≥ 4π (Qₑ)²

forces a rigid geometric structure in a neighborhood of Σ². In particular, the electric field E is everywhere normal to the foliation, and the local geometry splits isometrically as a Riemannian product. We then extend these rigidity phenomena to both compact and non-compact time-symmetric initial data sets with boundary, establishing sharp area–charge inequalities and analyzing the resulting rigidity of the boundary and ambient geometries under appropriate hypotheses, including cases with non-spherical boundary topology. As time permits, we conclude with several model examples illustrating these results. The second part of the lecture is based on joint work with T. Cruz (UFAL).

Title: Compact conformal gradient solitons

Abstract: In this lecture we will talk about gradient Yamabe solitons to the Yamabe flow and their generazations called conformal gradient solitons. Specifically, we will make an overview of the compact Yamabe solitons theory and to show the advances to the compact conformal gradient solitons case. 

This work is in partnership with Shun Maeta from Chiba University - Japan.

Title: Overdetermined Elliptic Problems on model Riemannian manifolds

Abstract:  In this talk, we present some rigidity theorems for annular sector-like domains in the setting of overdetermined elliptic problems on model Riemannian manifolds. Specifically, if such a domain admits a solution to the inhomogeneous Helmholtz equation satisfying both constant Dirichlet and constant Neumann boundary conditions, then the domain must be a spherical sector, and the solution must be radially symmetric.

Title: Classification of fractional Delaunay metrics on a twice-punctured sphere

Abstract: Fractional Delaunay metrics are the rotationally symmetric, periodic, complete metrics on the twice-punctured sphere with constant fractional Q_s-curvature, extending the classical Delaunay family from the scalar curvature case s = 1 to the nonlocal regime. In this talk, we present a full classification theorem: for every dimension n ≥ 3 and every fractional order s sufficiently close to one, any complete, conformally flat, constant Q_s-curvature metric on S^n \ {p, q} must be a member of the Delaunay family. The proof combines the Emden–Fowler reduction with a sharp comparison argument centered on the spherical bubble. The key input is that the bubble has a Morse index equal to one, which forces any solution whose maximum is normalized at t = 0 to coincide with a unique Delaunay profile. This collapses the moduli space to the necksize parameter and rules out all other possible behaviors. We further show that all fractional Delaunay metrics are nondegenerate for s close to one. These results provide a complete description of constant fractional Q_s-curvature metrics with two isolated singularities near the classical (s = 1) case.

Title: On quasi-Einstein manifolds with constant scalar curvature

Abstract: In this presentation, we study quasi-Einstein manifolds with constant scalar curvature. We discuss compact and noncompact (possibly with boundary) quasi-Einstein manifolds with constant scalar curvature R and prove a complete classification of 3-dimensional m-quasi-Einstein manifolds with constant R. Furthermore, we classify T-flat quasi-Einstein metrics, where T-tensor is directly related to the Cotton and Weyl tensors. In addition, we provide a classification in the compact case in dimension 4,  with constant scalar curvature.

Title: Geometric flows on homogeneous spaces

Abstract: In this talk, I will survey several results concerning the Ricci flow of invariant metrics (called the Homogeneous Ricci Flow) on a class of homogeneous manifolds known as generalized flag manifolds.

Title: Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

Abstract: Let (Σ², ds²) be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form L = Δ + V − aK together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential W and (in the curvature-corrected setting) the geodesic curvature κ_g of ∂Σ. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of Σ and the integrals of V and W, obtained via a Hersch balancing argument on the capped surface. As a geometric application, we derive sharp topological restrictions for compact two-sided free boundary minimal surfaces of Morse index at most one inside geodesic balls of negatively curved pinched Cartan–Hadamard 3-manifolds under a mild radius condition.

Based on a joint work with Marcos P. Cavalcante and Vinicius Souza.