We study the hyperbolicity of metric spaces in the Gromov sense. We deduce the hyperbolicity of a space from the hyperbolicity
of its “building block components”. These results are valuable since they simplify notably the topology of the space and allow
to obtain global results from local information. We also study how the punctures and the decomposition of a Riemann surface
in Y-pieces and funnels affect the hyperbolicity of the surface.
The Hodge–de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.