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 \(\mathbb{H}^{3}(-1)\).
We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometricall...
We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal su...
B. Firester, Raphael Tsiamis, Zi-Hui Zhao· 0 citations
For maps from surfaces, harmonicity is conformally invariant and a conformal immersion is harmonic precisely when its image is minimal. Biharmonicity is a fourth-order extension of this theory, but it is not conformally invariant. A nonminimal immersion may therefore become biharmonic after a suitable change of the dom...
We study a logarithmic re-gauging of the visual boundary, arising from the logarithmic metric on the sublinear Morse boundary introduced by Garg-Jana-Qing. For every proper geodesic hyperbolic space $X$, we prove that $d_{\mathrm{log}}(\xi,\eta)\asymp (1+(\xi\mid\eta)_o)^{-\theta}$. As a consequence, under the natural...
Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earl...
We study the deformation theory of harmonic maps with isolated singularities between compact Riemannian manifolds, in the case where all the tangent maps are smooth (away from the origin) and the decay to the tangents is polynomial. We will refer to these as conically singular harmonic maps. Under certain conditions on...
Dominik Gutwein, Thibault Langlais· 1 citation
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