R. Graham (Univ. of Washington): Gauss--Bonnet Formula for Renormalized Area of Minimal Submanifolds
Channel: Centrum Fizyki Teoretycznej PAN
Date: 2023-09-29

Robin Graham (University of Washington).
Gauss--Bonnet Formula for Renormalized Area of Minimal Submanifolds of Poincaré--Einstein Spaces.
Abstract:  I will describe the derivation of a formula of Gauss--Bonnet type involving the renormalized area of a minimal submanifold of a Poincaré--Einstein space.  This requires development of several new ingredients:  a special compactification coming from scattering theory on the minimal submanifold, conformally invariant powers of the Laplacian and Q-curvature on submanifolds of conformal manifolds, and a conjectured analog of a celebrated result of Alexakis asserting a decomposition of integrands of conformally invariant integrals.  This is joint work with Jeffrey Case, Tzu-Mo Kuo, Aaron Tyrrell, and Andrew Waldron.

This tak was given as a part of a a bi-weekly seminar for the Grieg collaboration project SCREAM: Symmetry, Curvature Reduction, and EquivAlence Methods, funded by the Norwegian Financial Mechanism 2014-2021. The registration number of the project is 2019/34/H/ST1/00636. 

The focus of the project is on a broad class of differential geometric structures known as Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2, 3, 5)-distributions, parabolic contact structures, and many more besides. Moreover, we explore the interaction of these structures with a variety of subjects such as mechanical systems and geometric robots, the theory of integrable systems, and Penrose's Conformal Cyclic Cosmology programme. 

More information can be found at www.cft.edu.pl/grieg.