Curvature surfaces in generic conformally flat hypersurfaces arising from Poincaré metric -- Extension and Approximation

Abstract

We study generic conformally flat (analytic-)hypersurfaces in the Euclidean $4$-space ${\mathbb{R}}^4$. Such a local-hypersurface is obtained as an evolution of surfaces issuing from a certain surface in ${\mathbb{R}}^4$, and then, in consequence, the original surface is a (principal-)curvature surface of the hypersurface. The Poincaré metric ${\check{g}}_H$ of the upper half plane leads to a $6$-dimensional set of rational Riemannian metrics $g_0$ of ${\mathbb{R}}^2$: on a simply connected open set in the regular domain of $g_0$, a curvature surface $f^0$ with the metric $g_0$ is determined, which we denote by $(f^0,g_0)$. In this paper, we choose a suitable metric $g_0$ of ${\mathbb{R}}^2$ determined by ${\check{g}}_H$ to get nice curvature surfaces (but it also has degenerate and divergent points in ${\mathbb{R}}^2$), and clarify the structure of the curvature surfaces $(f^0,g_0)$: the curvature surfaces $(f^0,g_0)$ extend analytically to what kind of set in ${\mathbb{R}}^2$ beyond the regular set of $g_0$, and then the extended surface $(f^0,g_0)$ is defined on a certain open set of ${\mathbb{R}}^2$ and bounded in ${\mathbb{R}}^4$; for the extended surface $(f^0,g_0)$, we explicitly catch the set of degenerate points and the limits in ${\mathbb{R}}^4$ of both ends of every principal curvature line, and then the two limits of every line for one principal curvature are parallel small circles in a standard $2$-sphere ${\mathbb{S}}^2$. Then, every principal curvature line in the extended surface $(f^0,g_0)$ is expressed by a frame field of ${\mathbb{R}}^4$ induced on the surface from a hypersurface and it lies on a standard $2$-sphere ${\mathbb{S}}^2$ with line-dependent radius. We also provide a general method of constructing an approximation of such frame fields, and obtain the entire pictures of those lines including degenerate points of $(f^0,g_0)$.