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}}<{\t...