Gauss Maps in Hyperbolic Surface Theory:A Unified Perspective

Immersed surfaces in hyperbolic three-space carry several natural Gauss-type maps with distinct geometric roles. The hyperbolic Gauss maps record the ideal endpoints of oriented normal geodesics; the Legendre Gauss lift retains the position-normal data and its contact structure; adjusted Gauss maps arise from gauge normalization and Iwasawa splitting in Weierstrass--Kenmotsu representations; and the conformal Gauss map encodes the mean-curvature sphere congruence in M\"obius geometry. We present these constructions in a common framework, emphasizing their target spaces, analytic properties, and mutual relations. Particular attention is given to the generalized DPW method, the necessity of flatness in adjusted rank-one data, and the harmonic-map characterization of Willmore surfaces. The resulting viewpoint distinguishes the asymptotic, contact, integrable, and conformal information carried by an immersed surface in $\mathbb{H}^{3}(-1)$.