Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Abstract

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.