Curvature surfaces in generic conformally flat hypersurfaces arising from Poincar\'{e} metric -- Extension and Approximation

TL;DR

This paper investigates the structure and extension of curvature surfaces in conformally flat hypersurfaces in our-dimensional space, using Poincare9 metrics to explicitly describe principal curvature lines and their limits.

Contribution

It introduces a method to extend and analyze curvature surfaces derived from Poincare9 metrics, detailing their boundary behavior and principal curvature lines in our-space.

Findings

Extended curvature surfaces are bounded and analytically extendable beyond regular sets.

Principal curvature lines approach small circles on or a sphere with line-dependent radius.

A general approximation method for frame fields on these surfaces is provided.

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 Poincar\'{e} 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 $(f^0,g_0)$.