Graduate Student Seminar, 4:30pm February 17, 2003, Hayes-Healy 229


Ion Dinca


Families of Surfaces


The Gauss-Codazzi equations give a complete description of a surface, that is the first and second fundamental forms uniquely describe the surface, modulo isometries of the ambient three dimensional euclidean space (rotations and translations). When put in complex conformal coordinates, these equations take a simple form, therefore we use these equations to describe them. They are the compatibility equations U_{z}-V_{\bar z}-[U,V]=0, equation which appear in soliton theory. Similar to soliton theory, such an equation is associated to a family of equations, and with a family of solutions.

This can also be seen since we have three real equations with 4 unknowns and therefore one can expect sometimes a 1 parameter family of solutions. Some examples are given, including Bonnet families (the metric and mean curvature are preserved), Amsler surfaces (negative curvature -1 and 2 asymptotic lines) related to the sine-Gordon equation and constant mean curvature tori, related to the elliptic sinh-Gordon equation.

To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert