From Notices of the AMS
Colding-Minicozzi Entropy and Complexity of Submanifolds
by Jacob Bernstein
Lu Wang
Communicated by Chikako Mese
Introduction
Intuitively, a sea urchin is more complicated than a billiard ball. However, it is not so easy to see how to formalize and quantify this distinction, i.e., what constitute good measures of the complexity of geometric objects. There are many different approaches, but on a high level, satisfactory answers should all involve something with the following properties:
1. Respects the geometric symmetries;
2. Collections of objects, taken together, should be more complex than the constituent elements;
3. The geometrically "simplest" elements in natural subclasses should be extrema of the quantity.
In addition to these general properties, it proves helpful to add a more technical and specific property:
4. Monotonic along a geometric heat flow.
This is justified by the fact that such flows tend to simplify the geometry of the objects being evolved.
In this article we focus on the geometry of submanifolds of Euclidean space and the measure of complexity we introduce will be associated with the mean curvature flow.
- Also in Notices
- On the Theory of Anisotropic Minimal Surfaces
- Uncovering Data Across Continua: An Introduction to Functional Data Analysis
