Weingarten Type Surfaces in ℍ²× ℝ and 𝕊²× ℝ

Abstract

Abstract. In this article, we study complete surfaces Σ, isometrically immersed in the product space H × R or S × R having positive extrinsic curvature K e . Let K i denote the intrinsic curvature of Σ. Assume that the equation aK i + bK e = c holds for some real constants a = , b > and c. e main result of this article state that when such a surface is a topological sphere it is rotational.

Elliptic Weingarten Hypersurfaces of Riemannian Products

“…Also, it follows from (9) that k n = . Thus, the principal curvatures of the (f s , φ)-graph Σ at (f s (p), φ(s)) ∈ Σ are (11)…”

“…It follows from (11) that, if F := {f s : M 0 → M ; s ∈ I} is isoparametric and Σ is an (f s , φ)-graph in M × R, then all principal curvatures k i of Σ at any point (f s (p), φ(s)) are functions of s alone.…”

“…Remark 1. The main feature of elliptic Weingarten hypersurfaces is that the Hopf Maximum Principle applies to the strictly convex ones (see [11] for a detailed discussion in the case n = 2). In fact, it also applies to a certain class of non convex elliptic Weingarten hypersurfaces, provided that the monotonicity condition ( 14) is satisfied not only on the positive cone Γ + , but also on a cone Γ which contains Γ + properly (cf.…”

“…The following lemma, which plays a fundamental role here, characterizes elliptic Weingarten (f s , φ)-graphs (with {f s } isoparametric) as those whose associatedfunctions are solutions of a certain first order ordinary differential equation. In fact, it follows directly from the definition of Weingarten hypersurface, the relations (11), and equality (13).…”

“…Elliptic Weingarten surfaces (sometimes called special Weingarten surfaces) of R 3 and other three-spaces have been considered in many works [3,11,14,17].…”