On the asymptotic geometry of finite-type $k$-surfaces in three-dimensional hyperbolic space

Abstract: For $0<k<1$, a finite-type $k$-surface in $3$-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In [32], we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type $k$-surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of its end-points is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of M\"obius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type $k$-surfaces. The first, which will be called the generalized volume, is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy, is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrisations of the strata of the space of finite-type $k$-surfaces by open complex manifolds, we prove a new Schl\"afli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, M\"obius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schl\"afli-type formula.