Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems

TL;DR

This paper investigates the existence and uniqueness of periodic pseudospherical surfaces derived from well-posed Cauchy problems of the Novikov equation, establishing geometric structures and regularity results.

Contribution

It demonstrates the existence of exactly two families of periodic forms satisfying structural equations, and constructs compatible second fundamental forms using Kato's semi-group approach.

Findings

Existence of two unique families of periodic forms

Construction of universal connection forms

Well-posedness and $C^1$ regularity of solutions

Abstract

We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic initial datum, then there exists exactly two families of periodic one-forms satisfying the structural equations for a surface. Each pair then defines a metric of constant Gaussian curvature and a corresponding Levi-Civita connection form. We prove the existence of universal connection forms giving rise to second fundamental forms compatible with the metric. The main tool to prove our geometrical results is the Kato's semi-group approach, which is used to establish well-posedness of solutions of the Cauchy problem involved and ensure $C^1$ regularity for the first fundamental form and the Levi-Civita connection form.