Asymptotic behavior of flat surfaces in hyperbolic 3-space (Q841226)

The paper is devoted to the studying of the asymptotic behaviour of flat surfaces in hyperbolic three-space. By definition, an immersion \(f: D^*\to H^3\) of the unit punctured disc \(D^*=\{z\in C\mid 0<| z| <1 \}\) into the hyperbolic space \(H^3\), is called an end of a flat surface (a flat end), if its Gauss curvature vanishes everywhere. The end is referred to as complete, if \(f\) is complete at the origin with respect to the induced Riemannian metric. The end is referred to as regular, if the hyperbolic Gauss maps generated by \(f\) can be extended smoothly across \(z=0\), otherwise \(f\) is called an irregular end. A flat end generates a family of parallel flat surfaces, flat wave fronts, which admit some kind of singularities. The authors introduce an original geometric quantity, the pitch, connected with the hyperbolic Gauss maps, and apply it to prove the following statement. Theorem A. Suppose that the flat surface \(f\) is a complete regular end. Then for a sufficiently small \(\varepsilon>0\), the image \(f(D^*_\varepsilon)\) of \(D^*=\{z\in C\mid 0<| z| <\varepsilon \}\) is congruent to a portion of the image of \((t,h)\in[0,2\pi)\times (0,h_0)\to (\varphi_h(t),h)\in\mathbb R^3_+\) with \[ \varphi_h(t) = C e^{imt} h^{1+p} + o(h^{1+p}) \] for a non-zero constant \(C\), a constant \(p\geq 0\), and an integer \(m>0\). Here \(H^3\) is viewed in the standard Poincare upper half-space model, \(p\) is just the pitch of \(f\), \(m\) is the multiplicity of the end. Thus each complete regular flat end is asymptotic to an \(m\)-covering of a rotationally symmetric end (this is essentially the same as a Galvez-Martinez-Milan theorem, cf. [\textit{J. A. Gálvez, A. Martínez} and \textit{F. Milán}, Math. Ann. 316, No.~3, 419--435 (2000; Zbl 1003.53047)]). Moreover, the authors obtain some refinements of Theorem A by calculating the second term in the asymptotic expansion. The main result deals with the asymptotic behavior of \textsl{incomplete} ends. Theorem B. Suppose that the flat front \(f\) is an incomplete end, which is regular weakly complete of finite type. Suppose that it is not contained in a geodesic line in \(H^3\). Then for a sufficiently small \(\varepsilon>0\), the image \(f(D^*_\varepsilon)\) is congruent to a portion of the image of \((t,h)\in[0,2\pi)\times (0,h_0)\to (\varphi_h(t),h)\in \mathbb R^3_+\) with \[ \varphi_h(t) = \left( \frac{m+n}{m}\;e^{i(m-n)t} + \frac{m-n}{m}\;e^{i(m+n)t} \right) h^{1+p} + o(h^{1+p}), \quad p = \frac{n}{m}\in (0,1)\cup (1,\infty). \] So, the pitch \(p\) of an incomplete weakly complete regular end is a positive rational number not equal to 1. As a corollary of Theorems A and B, the authors prove the following classifying statement. Corollary C. The pitch \(p\) of a complete regular end takes its value in \((-1,0]\). The pitch \(p\) of an incomplete regular end, which is weakly complete of finite type, takes its value in \(Q_+\setminus\{ 1\}\). Moreover, a regular weakly complete end of finite type is {\parindent=4mm \begin{itemize}\item[{\(\cdot\)}] a snowman-type end if and only if \(-1<p<-1/2\); \item[{\(\cdot\)}] an horospherical end if and only if \(p=-1/2\); \item[{\(\cdot\)}] an hourglass-type end if and only if \(-1/2<p<0\); \item[{\(\cdot\)}] a complete cylindrical end if and only if \(p=0\); \item[{\(\cdot\)}] an end of epicycloid-type with \(2n\) cusps and winding number \(m\) end if and only if \(p=n/m\in (0,1)\); \item[{\(\cdot\)}] an end of hypocycloid-type with \(2n\) cusps and winding number \(m\) end if and only if \(p=n/m\in (1,\infty)\). \end{itemize}} Each item in this statement is described analytically.