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Weingarten surfaces in R^3 and complete hypersurfaces with negative Ricci curvature in R^{n+1}


This research project is divided into two parts. In the first one we are interested in studying Weingarten surfaces, that is, surfaces in which there is a (possibly nonlinear) relationship between their principal curvatures. Almost all works in the literature deal with the linear case, however the techniques used in them do not apply to the nonlinear case. We will seek in our investigations to develop new tools for the study of such surfaces, which apply to both the linear and the nonlinear cases.The second part of the project is related to the following generalization of Efimov's theorem, conjectured by Reilly and Yau: "For any complete hypersurface with negative Ricci curvature in $R^{n+1}$ one has $\inf | Ric |=0$." Smyth and Xavier proved that this conjecture is true in the case $n=3$, and Chern that it is true in the class of entire graphs. Subsequently, F. Fontenele proved that in this class of hypersurfaces it holds the stronger result that $\inf |A|=0$. The purpose of this part of the project is to refine the ideas contained in F. Fontenele's work and extend the estimate $inf |A|=0$ for a class of hypersurfaces wider than that of entire graphs. (AU)

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