Zoll manifolds are Riemannian manifolds all of whose geodesics are closed and have the same length. Beyond the round sphere, nontrivial examples were constructed by Funk and Guillemin, initiating a rich line of research. In this talk, I introduce a free-boundary analogue of this notion. A compact Riemannian manifold with boundary is said to be Zoll with boundary if every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. I will show that such manifolds exhibit strong rigidity: all free-boundary geodesics have the same length and share the same Morse index. Using Morse index theory and algebraic topology, we obtain a complete geometric and topological classification. In particular, when the boundary is connected, the manifold is a tubular neighborhood of a closed embedded submanifold (the “soul”), and the boundary fibers over the soul either as a sphere bundle or as a nontrivial two-fold covering. This is joint work with Paolo Piccione and Roney Santos.

Let $M$ be a properly embedded, connected, complete surface in $\mathbb{R}^3$ with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively – which we will refer to as Weingarten equation. In this talk, we discuss how the symmetries of $C$ may induce symmetries of the whole $M$. When $M$ is contained in one of the two halfspaces determined by $C$, we give sufficient conditions for $M$ to inherit the symmetries of $C$. In particular, when $M$ is vertically cylindrically bounded, we get that $M$ is rotational if $C$ is a circle. In the case in which the Weingarten equation is linear, we give a sufficient condition for such a surface to be contained in a halfspace. Both results are generalizations of results of Rosenberg and Sa Earp, for constant mean curvature surfaces, to the Weingarten setting. In particular, our results also recover and generalize the constant mean curvature case.

El establecimiento de resultados tipo Talenti para la simetrización de la solución de una ecuación de Poisson, planteada sobre un dominio regular $D$ en una variedad Riemanniana $(M,g)$, se encuentra estrechamente vinculado a la existencia de una desigualdad isoperimétrica satisfecha por los dominios de la variedad. Esta relación ha sido explorada previamente, tanto en el contexto compacto como en el no-compacto. Hasta la fecha, el conocimiento sobre comparaciones en espacios de curvatura negativa es limitado, más allá de lo establecido por McDonald (donde se prueba una comparación tipo Talenti en espacios hiperbólicos). La dificultad radica en que, en espacios de curvatura negativa, no se ha establecido una desigualdad isoperimétrica de forma general. Presentaremos un esquema de la demostración de una serie de comparaciones tipo Talenti para variedades Riemannianas completas y no compactas que cumplen una de las siguientes condiciones: • Poseen una constante isoperimétrica positiva. • Su perfil isoperimétrico está controlado en cierto sentido. Estas hipótesis son integradoras en el sentido de que, en el caso no compacto, engloban los contextos ya estudiados e incluyen además a las variedades de Cartan-Hadamard. Trabajo en colaboración con V. Gimeno.

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