# Renormalized Area for Minimal Hypersurfaces of 5-Dimensional Poincaré-Einstein Spaces

# Renormalized Area for Minimal Hypersurfaces of 5-Dimensional Poincaré-Einstein Spaces

We will report on a result within the holographic study of conformal geometry initiated by Fefferman and Graham:

In 1999 Graham and Witten showed that one can define a notion of renormalized area for properly embedded minimal submanifolds of Poincaré-Einstein spaces. For even dimensional submanifolds, this quantity is a global invariant of the embedded submanifold. In 2008 Alexakis and Mazzeo wrote a paper on this quantity for surfaces in a 3-dimensional PE manifold, getting an explicit formula and studying its functional properties. We will look at a formula for the renormalized area of a minimal hypersurface of a 5-dimensional Poincaré-Einstein space in terms of a Chern-Gauss-Bonnet formula. We will show how the integrand in the renormalized area formula can be realized as a conformal integral hypersurface invariant.