LAUSR

In this paper, we study the quasi-local energy (QLE) and the surface geometry for Kerr spacetime in the Boyer-Lindquist coordinate without taking the slow rotation approximation. We also consider in the region $r r_{k}(a)$, the Gaussian curvature is positive of the surface with constant $t,r$, and for $r>\sqrt{3}a$ 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 $r=\sqrt{3}a$ intersect at the point $a=\sqrt{3}m/2$, which is the limit for the horizon to be isometrically embedded into $\mathbb{R}^3$ [18]. The numerical result indicates that the Kerr QLE is monotonically decreasing to ADM $m$ from the region inside the ergosphere to large $r$. For increasing $a$, the QLE is decreasing but for increasing irreducible mass $M_{\mathrm{ir}}$, QLE is increasing. From the results of Chen-Wang-Yau [7], we conclude that in a certain region $r>r_{h}(a)$, the critical value of the Kerr QLE is a global minimum.