Sesión Geometría y Topología

Diciembre 14, 15:30 ~ 15:50

Isoparametric hypersurfaces in the complex hyperbolic space

Domínguez Vázquez, Miguel

A hypersurface of a Riemannian manifold is called isoparametric if its sufficiently close equidistant hypersurfaces have constant mean curvature. These objects have been studied since the beginning of the 20th century and their investigation has given rise to a beautiful area of research over the last few decades. However, the classifications of isoparametric hypersurfaces in Euclidean and real hyperbolic spaces due to Beniamino Segre and Élie Cartan in the 30s have been, until very recently, the only ones of this kind known for a complete family of symmetric spaces. Some years ago, we discovered many examples of isoparametric hypersurfaces with nonconstant principal curvatures in complex hyperbolic spaces \cite{mz}. This phenomenon reveals an important difference (and difficulty) with respect to the case of space forms. Recently, in a joint work with José Carlos Díaz-Ramos and Víctor Sanmartín-López, we have completely classified isoparametric hypersurfaces in complex hyperbolic spaces~\cite{DDV:adv}. The aim of this talk is to present and explain this classification result. \begin{thebibliography}{11} \bibitem{mz} J.\ C.\ Díaz-Ramos, M.\ Domínguez-Vázquez: Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces. Math.\ Z.\ 271 (2012), 1037--1042. \bibitem{DDV:adv} J.\ C.\ Díaz-Ramos, M.\ Domínguez-Vázquez, V. Sanmartín-López: Isoparametric hypersurfaces in complex hyperbolic spaces. Adv.\ Math.\ 314 (2017), 756--805. \end{thebibliography}

Autores: Domínguez Vázquez, Miguel.