EVENT DETAILS AND ABSTRACT

Title: The area preserving curve shortening flow in a free boundary setting
Speaker: Elena Maeder-Baumdicker
Speaker Info: Karlsruhe Institute of Technology (KIT)/Princeton
Brief Description:
Special Note:
Abstract:

Under the area preserving curve shortening flow (APCSF), a convex simple closed plane curve converges smoothly to a circle with the same enclosed area as the initial curve (Gage 1986). Note that an embedded circle is the solution of the isoperimetric problem in $\mathbb R^2$. Corresponding to the outer isoperimetric problem for a convex domain I will present results concerning the APCSF with Neumann free boundary conditions outside of a convex domain. Under certain conditions on the initial curve the flow does not develop a singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given domain and enclosing the same area as the initial curve. I will, on the other hand, explain that there are many initial curves developing a singularity in finite time. In all these cases, the singularity is of type II, and I conjecture that some curves developing a singularity stay embedded under the flow.