Friday February 10th, 2023 | |
| Time: | 11:00 AM - 12:00 PM |
| Title: | A Proof that CAT(k) Surfaces Have Bounded Integral Curvature |
| Speaker: | Saajid Chowdhury, Hechen Hu, Adam Tsou, Stony Brook University |
| Location: | Math P-131 |
| Abstract: | |
| The field of Alexandrov Geometry studies the geometric properties of metric spaces where properties such as angle are well-defined. In this talk, we consider CAT(κ) surfaces. These are defined as surfaces in which small geodesic triangles are thinner than triangles drawn with matching edge lengths in a surface of constant curvature. This condition gives an upper curvature bound on the surface. A separate notion of curvature bound on a surface is to have a uniform upper bound on the angle excess of any finite collection of non-overlapping triangles contained in a given compact neighborhood. This condition is known as having bounded integral curvature. We give a proof of a folklore theorem that a CAT(κ) surface is also a surface of bounded integral curvature. Our proof uses a new tool called vertex-edge triangulations, which are triangulations obtained by repeatedly subdividing a base triangulation. Their usefulness comes from the observation that angle excess and model area of a triangulation behave monotonically under subdivision. By a Gauss–Bonnet type argument, we derive a uniform upper bound on angle excess of a vertex-edge triangulation refining an arbitrary finite collection of triangles. This research was carried out as part of the 2022 Stony Brook REU program. | |