Motonicity theorems for minimal surfaces and linkedness of their boundary.

02.11. bis 02.11.2021, 12:15-13:45  –  Room 2.09.2.22 Campus Golm, C9A03 Tübingen
Geometric Analysis, Differential Geometry and Relativity

Manh Tien Nguyen

I will explain how each function whose Hessian is a multiple of the metric of a Riemannian manifold $M$ corresponds to a monotonicity theorem for minimal surfaces in $M$. When $M$ is the hyperbolic space, such functions arise as the Minkowskian coordinates in the hyperboloid model and they pose constraints on where a minimal surface can pass by in terms of its boundary curve. Using these constraints, one can detect the linkedness of a link in S^3 by counting the number of minimal surfaces in H^4 filling it. If time allows, I will present two different upper bounds of the Graham--Witten's renormalised area obtained from the monotonicity theorems, one by the time coordinate and one by the space coordinate.