Non-positively curved Ricci Surfaces with catenoidal ends

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