In this talk, we will explore the conditions for the existence or non-existence of non-trivial Einstein multiply warped products (MWP), with a particular focus on generalized Kasner-type manifolds. Starting from the definition of warped products between semi-Riemannian manifolds, we will discuss how the warping function affects the geometry of Einstein metrics. We will present examples illustrating the absence of non-trivial Einstein MWP in both general and generalized Kasner settings, as well as provide estimates of the Einstein parameter that govern their existence. In particular, we will present results that constrain the range of admissible proportionality constants between the Ricci tensor and the metric tensor in compact Riemannian manifolds with positive scalar curvature.
The k-Yamabe flow is a fully no-linear extension to the Yamabe flow that appears nat- urally in problems related to topological classification in higher dimensions. In this talk we describe the construction, classification and asymptotic behavior of radially symmetric gra- dient k-Yamabe solitons that are locally conformally flat [ES24], these are special solutions to this flow that play a central role in the theory. Our study extends the results obtained by P. Daskalopouolos and N. Sesum in [DS13] in the case n > 2k. This is joing work with Mariel Sáez.
References
[DS13] Panagiota Daskalopoulos and Natasa Sesum. The classification of locally conformally flat yamabe solitons. Advances in Mathematics, 240:346–369, 2013.
[ES24] María Fernanda Espinal and Mariel S áez. On the existence and classification of k-yamabe gradient solitons. arXiv preprint arXiv:2410.06942, 2024.
In this talk, we consider the mass-constrained minimization of interaction energies having two competing terms. A long-range repulsion, and a short-range attraction. In particular, we are interested in study (local) minimality properties of shapes.
The two competing terms in the first energy are both isotropic. We will thus study the stability properties of the ball.
The second has an anisotropic repulsive term. In two and three spatial dimension, we will be able to determine the global minimizer.
This talk is based on works in collaboration with Marco Bonacini (Università di Trento), Maria Giovanna Mora (Università di Pavia), Lucia Scardia (Heriot-Watt University), and Ihsan Topaloglu (Virginia Commonwealth University).
We shall discuss a shape sensitivity result, obtained in collaboration with Luis González (UACH), for the equilibrium
energy, in the Oseen-Frank model, of a nematic liquid crystal under normal anchoring conditions on the boundary. As an application we show that, for a nematic liquid crystal confined between two parallel cylindrical domains, the coaxial configuration is a critical state for the equilibrium energy, partially resolving a conjecture made by Alouges and Coleman.
In this talk I will present a general scheme to construct a global, Delaunay-type solution to the Bernoulli one-phase free boundary problem in dimension $(n+1)$ that is periodic in the last variable, starting from a global solution in dimension $n\geq 2$ that has finite Morse index. As a result, we construct a new solution to the one-phase
free boundary problem in dimension 3, arising from the double hairpin solution of Hauwirth-Hélein-Pacard in R^2.
We discuss the regularity properties of two-dimensional stable s-minimal surfaces, presenting a robust estimate and an optimal sheet separation bound, according to which the distance between different connected components of the surface must be at least the square root of 1-s.
Let $(M^n, [\hat{g}])$ be the conformal infinity of an asymptotically hyperbolic Einstein (AHE) manifold $(X^{n+1},g^+).$ We will take the scattering operator associated to the AHE filling in as the fractional conformal Laplacian. Equipped with fractional conformal Laplacians defined via the AHE manifold, we can define a fractional Yamabe problem, looking for a conformal metric of $(M^n,[\hat{g}])$ which has constant fractional scalar curvature. We will present some new developments on the fractional Yamabe problem assuming an AHE filling in.
The mean curvature flow is one of the most studied geometric flows, arising naturally as the gradient flow of the area functional for hypersurfaces in a Riemannian manifold. Its stationary solutions correspond to minimal hypersurfaces, which have been extensively studied through variational methods and which have been the object of significant progress in the last decade, notably in the context of Morse theory and Yau's conjecture on the existence of infinitely many minimal hypersurfaces—now resolved through groundbreaking work by F. Marques, A. Neves, A. Song, X. Zhou, among others.
In this talk, we discuss some existence and rigidity problems for mean curvature flows in compact Riemannian manifolds. We also explore questions concerning the volume spectrum, a sequence of critical values for the area functional produced via min-max techniques and which shares intriguing properties with the spectrum of the Laplacian. In particular, we will discuss how the phase transitions approximation for the area functional, using the Allen-Cahn equation, has provided new insights and raised novel questions in geometric analysis.
Given the complexity of the Einstein equations, it is often a good choice to study a question of interest in the framework of a restricted class of solutions. One way to impose such restrictions is to consider solutions that satisfy a given symmetry condition. In this work, we consider the particular class of spacetimes that admit two space-like Killing vector fields. More precisely, we will focus on the Einstein vacuum model $R_{\mu \nu}(\tilde g)=0$, where $\tilde g$ is the metric tensor and $R_{\mu \nu}$ is the Ricci tensor, in the Belinski-Zakharov setting. This ansatz is compatible with the well-known Gowdy symmetry. The main goal of this talk is to describe rigorously the conditions for the global existence of small solutions, and their decay in the light cone, as well as the stability of a first set of solitonic solutions (gravisolitons), for the so-called reduced Einstein equation, viewed as an identification of the Principal Chiral Field (PCF) model.