ar X iv : d g - ga / 9 61 00 06 v 1 9 O ct 1 99 6 BONNET PAIRS AND ISOTHERMIC SURFACES

A classical question in surface theory is which data are sufficient to describe a surface in space up to rigid motions. Bonnet suggested that mean curvature and metric should suffice to determine the surface generically. The local theory was developed by Bonnet [6], Cartan [4] and Chern [5] who showed the existence of various 1-parameter families of Bonnet surfaces, i.e., surfaces with the same induced metric and mean curvature. A comprehensive study of this problem and its relationship to the Painleve equations has recently been completed by Bobenko and Eitner [2]. On the other hand, Lawson and Tribuzy [9] have shown that for embedded compact surfaces there are at most two surfaces to a given metric and mean curvature. Moreover, uniqueness can be established under various global assumptions [8]. Up to date it is unknown whether such compact Bonnet pairs exist and if, how to construct them.