We consider hypersurfaces of products \(M\times \mathbb{R}\) with constant r-th mean curvature — to be called \(H_r\)-hypersurfaces — where \(M\) is an arbitrary Riemannian manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds \(M\), including all simply connected space forms and the Hadamard manifolds known as Damek-Ricci spaces. Uniqueness results for complete \(H_r\)-hypersurface of \(\mathbb{H}^n\times\mathbb{R}\) or \(\mathbb{S}^n\times\mathbb{R}\) \((n \geq 3)\) are also obtained. This is a joint work with Ronaldo de Lima (UFRN) and Fernando Manfio (ICMC-USP).

Sala EINSTEIN UGR (virtual)

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In this talk, we will describe the 1-parameter family of horizontal Delaunay surfaces in \(\mathbb{S}^2\times\mathbb{R}\) and \(\mathbb{H}^2\times\mathbb{R}\) with supercritical constant mean curvature. These surfaces are not equivariant but singly periodic, and they lie at bounded distance from a horizontal geodesic. We will show that horizontal unduloids are properly embedded surfaces in \(\mathbb H^2\times\mathbb{R}\). We also describe the first non-trivial examples of embedded constant mean curvature tori in \(\mathbb S^2\times\mathbb{R}\) which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. They have constant mean curvature \(H>\frac{1}{2}\). Finally, we prove that there are no properly immersed surface with critical or subcritical constant mean curvature at bounded distance from a horizontal geodesic in \(\mathbb{H}^2\times\mathbb{R}\).

Acceso a sala Zoom.

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We consider the following critical \(p\)-Laplace equation: \begin{equation}(1)\quad \Delta_p u+u^{p^{\ast}-1}=0 \quad \text{ in \(\mathbb{R}^n\)} \, , \end{equation} with \(n \geq 2\) and \(1 < p < n\). Equation (1) has been largely studied in the PDE's and geometric analysis' communities, since extremals of Sobolev inequality solve (1) and, for \(p=2\), the equation is related to the Yamabe's problem. In particular, it has been recently shown, exploiting the moving planes method, that positive solutions to (1) such that \(u\in L^{p^\ast}(\mathbb{R}^n)\) and \(\nabla u\in L^p(\mathbb{R}^n)\) can be completely classified. In the talk we will consider the anisotropic critical \(p\)-Laplace equation in convex cones of \(\mathbb{R}^n\). Since the moving plane method strongly relies on the symmetries of the equation and of the domain, in the talk a different approach to this problem will be presented. In particular this approach gives a complete classification of the solutions in an anisotropic setting. More precisely, we characterize solutions to the critical \(p\)-Laplace equation induced by a smooth norm inside any convex cone of \(\mathbb{R}^n\).
This is a joint work with G. Ciraolo and A. Figalli.

In this lecture, we are going to talk about a version of the barrier principle for varifolds at infinity. The main aims of this lecture is to prove the validity of the equality

\( \mathrm{dist}(\partial\Omega,\mathrm{spt}\|\Sigma\| )= \mathrm{dist}(\partial\Omega,\mathrm{spt}\|\partial\Sigma\| ), \)

when \(\Omega\) is an open set in a complete Riemannian manifold \(M\) both with a particular structure and \(\Sigma\) is varifolds with bounded mean curvature satisfies a particular condition. This work was done jointly with Jorge H. de Lira (Universidad Federal do Ceará), Luciano Mari (Universitá degli Studi di Torino) and Adriano A. de Medeiros (Universidade Federal da Paraı́ba).

Acceso a la sala

We will discuss a type of semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically flat manifolds. In particular, electromagnetic fields give rise to this kind of systems. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on asymptotically flat manifolds with boundary, which extends and clarifies previously known results.

It is important to understand Lagrangian self-shrinkers with simple geometry since it is the starting point of singularity analysis for the Lagrangian mean curvature flow. One interesting observation is that all known embedded examples in \(\mathbb{R}^4\) become the Clifford Torus. Hence it is natural to ask whether the Clifford Torus is unique as an embedded Lagrangian self-shrinker in \(\mathbb{R}^4\). In this direction, we recently proved that a closed Lagrangian self-shrinker in \(\mathbb{R}^4\) symmetric with respect to a hyperplane is given by the product of two Abresch-Langer curves and obtained a positive answer for the question by assuming reflection symmetry. In this talk, we will focus on the motivation for this work and the reason why reflection symmetry was assumed. Moreover, the idea of proof will also be discussed.

Link:
https://oficinavirtual.ugr.es/redes/SOR/SALVEUGR/accesosala.jsp?IDSALA=22968085
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In this lecture, I will outline the technique of integrable systems for CMC surfaces, but with a view at some other cases. Then I will explain some recent developments in the construction of certain families of CMC surfaces using this setup. In particular, we start with a \(2\times 2\) Cauchy problem to which we associate a scalar second-order differential equation. The singularities in this ODE correspond to the ends in the resulting surface. Particularly, regular singularities produce asymptotically Delaunay ends while irregular singularities produce irregular ends. Our aim is to discuss global issues such as period problems and asymptotic behavior involved in the construction of CMC surfaces in \(\mathbb{R}^3\) arising from the family of Heun's differential equations.

Link to the Zoom meeting
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One of the new four-dimensional gauge theories is a set of equations discovered by Kapustin and Witten in 2005. This theory involves a boundary condition which in turn involves a knot K in a 3-manifold. A later conjecture by Gaiotto and Witten states that a sequence of numerical counts of elements of the moduli space of solutions determines the Jones polynomial of the knot. While this remains open, quite a lot has been discovered. I will briefly survey these developments, including the basic structural and regularity theory of these equations, which was joint work with Witten, and a fairly complete resolution of this problem in the dimensionally reduced case, on the product of a Riemann surface and a half-line, which was joint work with S. He.

Acceso a la sala; access to the room
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The network flow is a system of parabolic differential equations that describes the motion of a family of curves in which each of them evolves under curve-shortening flow. This problem arises naturally in physical phenomena and its solutions present a rich variety of behaviors. The goal of this talk is to describe some properties of this geometric flow and to discuss an alternative proof of short-time existence for non-regular initial conditions. The methods of our proof are based on techniques of geometric microlocal analysis that have been used to understand parabolic problems on spaces with conic singularities. This is joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda.

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Password of the virtual room: 808148

The Gross--Pitaevskii equation is a nonlinear Schrödinger equation that models the behavior of a Bose-Einstein condensate. The quantum vortices of the condensate are defined by the zero set of the wave function at time \(t\). In this talk we will present recent work about how these quantum vortices can break and reconnect in arbitrarily complicated ways. As observed in the physics literature, the distance between the vortices near the breakdown time, say \(t = 0\), scales like the square root of $t$: it is the so-called \(t^{1/2}\) law. At the heart of the proof -- which ultimately entails understanding the evolution of curves in space -- lies a remarkable global approximation property for the linear Schrödinger equation. The talk is based on joint work with Daniel Peralta-Salas.
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THIS SEMINAR HAS BEEN POSTPONED BECAUSE OF THE UNIVERSITY POLICY CONCERNING THE CORONAVIRUS OUTBREAK. A NEW DATE WILL BE ANNOUNCED AS SOON AS POSSIBLE.

In this lecture I will outline the integrable systems technique for CMC surfaces, but with a view at some other cases. Then I will explain some recent developments in the construction of certain families of CMC surfaces using this setup. In particular, we start with a \(2\times 2\) Cauchy problem to which we associate a scalar second order differential equation. The singularities in this ODE correspond to the ends in the resulting surface. Particularly, regular singularities produce asymptotically Delaunay ends while irregular singularities produce irregular ends. Our aim is to discuss global issues such as period problems and asymptotic behavior involved in the construction of CMC surfaces in \(\mathbb{R}^3\) arising from the family of Heun's differential equations.

We say that a surface is semigraphical if it is properly embedded, and, after removing a discrete collection of vertical lines, it is a graph. In this talk, we provide a nearly complete classification of semigraphical translators.

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quantum field theory and black hole evaporation.

I will discuss the subject of real Kaehler submanifolds, that is, isometric immersions \(f\colon M^{2n}\to\mathbb{R}^{2n+p}\) of a Kaehler manifold \((M^{2n},J)\) of complex dimension \(n\geq 2\) into Euclidean space with codimension \(p\). In particular, I will present a recent result that shows that for codimension \(2p\leq 2n-1\) generic rank conditions on the second fundamental form of \(f\) imply that the submanifold has to be minimal. In fact, for codimension \(p\leq 11\) we have a stronger conclusion, namely, that \(f\) must be holomorphic with respect to some complex structure in the ambient space.