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By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic 3 3 -manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic 3 3 -manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one. 

Abstract We show that the bounded Borel class of any dense representation $$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $$G$$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$$ is uniformly separated in semi-norm from any other representation $$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$$ for which there is a subgroup $$H\le G$$ on which $$\rho $$ is still dense but $$\rho ^{\prime}$$ is discrete or indiscrete but stabilizes a point, line, or plane in $${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of $${\operatorname{PSL}}_2{\mathbb{R}}$$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.