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

#### Speaker:

Ion Dinca
#### Title:

Families of Surfaces
#### Abstract:

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