Gap theorems for linear Weingarten surfaces

Abstract

We deal with complete surfaces in a 3-dimensional Riemannian space form $${\mathbb {Q}}^3_c$$ of constant sectional curvature $$c\in \{-1,0,1\}$$, satisfying a linear Weingarten relation of the type $$K=aH+b$$, where a and b are constant and H and K denote the mean and Gaussian curvatures, respectively. Under suitable constraints on the values of the constants a and b, we obtain gap theorems showing that such a surface must be either totally umbilical or isometric to a hyperbolic cylinder $${\mathbb {H}}^{1}(-\sqrt{1+r^{2}})\times {\mathbb {S}}^{1}(r)$$, when $$c=-1$$, a circular cylinder $${\mathbb {R}}\times {\mathbb {S}}^{1}(r)$$, when $$c=0$$, and a flat torus $${\mathbb {S}}^{1}(\sqrt{1-r^{2}})\times {\mathbb {S}}^{1}(r)$$, when $$c=1$$. Our approach is based on a weak maximum principle for an appropriate Cheng-Yau modified operator.