Surfaces in $\mathbb E^3$ with $L_1$-pointwise 1-type Gauss map

Young Ho Kim; Nurett$\dot{\text i}$n Cenk Turgay

doi:10.4134/BKMS.2013.50.3.935

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(3): 935-949

Printed May 31, 2013

https://doi.org/10.4134/BKMS.2013.50.3.935

Surfaces in $\mathbb E^3$ with $L_1$-pointwise 1-type Gauss map

Young Ho Kim and Nurett$\dot{\text i}$n Cenk Turgay

Kyungpook National University, Istanbul Technical University

Abstract

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Abstract

In this paper, we study surfaces in $\mathbb E^3$ whose Gauss map $G$ satisfies the equation $\square G=f(G+C)$ for a smooth function $f$ and a constant vector $C$, where $\square$ stands for the Cheng-Yau operator. We focus on surfaces with constant Gaussian curvature, constant mean curvature and constant principal curvature with such a property. We obtain some classification and characterization theorems for these kinds of surfaces. Finally, we give a characterization of surfaces whose Gauss map $G$ satisfies the equation $\square G=\lambda(G+C)$ for a constant $\lambda$ and a constant vector $C$.

Keywords: Gauss map, $\square$-pointwise 1-type, Cheng-Yau operator

MSC numbers: 53B25, 53C40