4. Bonnet Surfaces in S ³ and H ³ and Surfaces with Harmonic Inverse Mean Curvature

Abstract

Similar to Chapter 3, one considers the Bonnet problem in S ³ and H ³. The local theory of Bonnet surfaces in S ³ and H ³ without critical points of the mean curvature function was developed in [Vo], [ChL]. It was proven that all Bonnet surfaces in S ³ are Weingarten surfaces, and a classification similar to Cartan’s classification of Bonnet surfaces in ℝ³ was obtained. The Gauss-Codazzi equations of Bonnet surfaces in S ³ reduce to an ordinary differential equation similar to (3-18):

$$ \left( {\frac{{H''\left( t \right)}} {{H'\left( t \right)}}} \right)^\prime - H\prime \left( t \right) = \left| Q \right|^2 \left( {2 - \frac{{H^2 \left( t \right) + C}} {{H\prime \left( t \right)}}} \right) $$

((4.1))

), with C > 0.