Quasilocal energy and surface geometry of Kerr spacetime

We study the quasilocal energy (QLE) and the surface geometry for Kerr spacetime in the Boyer-Lindquist coordinates without taking the slow rotation approximation. We also consider in the region r <= 2m, which is inside the ergosphere. For a certain region, r > r(k)(a), the Gaussian curvature of the surface with constant t, r is positive, and for r > root 3a the critical value of the QLE is positive. We found that the three curves: the outer horizon r = r(+) (a), r = r(k) (a) and r = root 3a intersect at the point a = root 3m/2, which is the limit for the horizon to be isometrically embedded into R-3. The numerical result indicates that the Kerr QLE is monotonically decreasing to the ADM m from the region inside the ergosphere to large r. Based on the second law of black hole dynamics, the QLE is increasing with respect to the irreducible mass M-ir. From the results of Chen-Wang-Yau, we conclude that in a certain region, r > r(h), the critical value of the Kerr QLE is a global minimum.