Qing Han, Notre Dame

Title: The isometric embedding of abstract surfaces in the 3-dim Euclidean space

Abstract: A surface in the 3-dim Euclidean space can be viewed as the image of a map from a planar domain to the 3-dim Euclidean space, at least locally. The standard metric in the Euclidean space induces a metric on the surface, which allows us to compute the lengths of curves on the surface and to compute the distance of any two points on the surface. For example, the distance of two points on a sphere is the length of the small arc on the great circle through these two points. The induced metric on the surface can be transformed to an abstract metric by the abovementioned map. Now, we consider the converse question. Given an abstract metric on a planar domain, can we find a surface in the 3-dim Euclidean space whose induced metric is the given abstract metric? This is the isometric embedding problem we will discuss. It started with a conjecture by Schlaefli in 1873 that this can always be achieved near any given point. This conjecture is widely open and there are only a few results under various conditions. The question can be reformulated in terms of partial differential equations. Despite the technical description, the underlying equation has a simple form. In this talk, I will give a historical account and explain why this equation is hard to solve. The talk is aimed at a general audience.