Non-positively curved Ricci Surfaces with catenoidal ends

Abstract

A Ricci surface is defined to be a Riemannian surface $({\varvec{M}},{\varvec{g}}_{\varvec{M}})$ whose Gauss curvature ${\varvec{K}}$ satisfies the differential equation ${\varvec{K}}\varvec{\Delta } {\varvec{K}} + {\varvec{g}}_{\varvec{M}}\left( {{\textbf {d}}{\varvec{K}}},{{\textbf {d}}{\varvec{K}}}\right) + {\textbf {4}}{\varvec{K}}^{\textbf {3}}={\textbf {0}}$. In the case where ${\varvec{K}}<{\textbf {0}}$, this equation is equivalent to the well-known Ricci condition for the existence of minimal immersions in ${\mathbb {R}}^3$. Recently, Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally an isometric minimal immersion into ${\mathbb {R}}^3$. In this paper, we are interested in studying non-compact orientable Ricci surfaces with non-positive Gauss curvature. Firstly, we give a definition of catenoidal end for non-positively curved Ricci surfaces. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data to obtain some classification results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. Furthermore, we also give an existence result for non-positively curved Ricci surfaces of arbitrary positive genus which have finite catenoidal ends.