Height estimate for special Weingarten surfaces of elliptic type in 𝕄²(𝕔)×ℝ

Abstract

In this article we provide a vertical height estimate for compact special Weingarten surfaces of elliptic type in M 2 ( c ) × R {\mathbb M}^2(c) \times \mathbb {R} , i.e. surfaces whose mean curvature H H and extrinsic Gauss curvature K e K_e satisfy H = f ( H 2 − K e ) H=f(H^2-K_e) with 4 x ( f ′ ( x ) ) 2 > 1 , 4x(f’(x))^2>1, for all x ∈ [ 0 , + ∞ ) . x \in [0,+\infty ). The vertical height estimate generalizes a result by Rosenberg and Sa Earp and applies only to surfaces verifying a height estimate condition. When c > 0 , c>0, using also a horizontal height estimate, we show a non-existence result for properly embedded Weingarten surfaces of elliptic type in H 2 ( c ) × R \mathbb {H}^2(c) \times \mathbb {R} with finite topology and one end.