Sobre o fluxo de curvatura no Plano Hiperbólico

Abstract

This work is based on the article ``{\it Soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space}" by da Silva and Tenenblat \cite{Ket}. Our goal is to present the proof that characterize when a regular curve is a soliton of the curvature flow. Namely, a regular curve $X : I \rightarrow\mathbb{H}^2$ parameterized by arc length is a soliton of the curvature flow, if only if, its geodesic curvature is equal to the pseudo inner product between its tangent vector field and a non-null vector of the Minkowski space. This result enables us to establish a relationship between the solitons and a system of ordinary differential equations. Through the system qualitative analysis, it is possible to proof that the solitons are defined curves on the entire real line, embedded in $\mathbb{H}^2$ and its geodesic curvature, at each end, converges to $-1$, $0$ ou $1$.