We study representations of lattices of $\mathrm{PU}(m,1)$ into $\mathrm{PU}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into $\mathrm{PU}(n,1)$ of non-uniform lattices in $\mathrm{PU}(1,1)$, and more generally of fundamental groups of orientable...

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov...