Some estimates of Wang-Yau quasilocal energy

Given a spacelike 2-surface $\Sigma$ in a spacetime $N$ and a constant future timelike unit vector $T_0 $ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(\Sigma, X, T_0)$ for a given isometric embedding $X$ of $\Sigma$ into a flat 3-slice in $\R^{3,1}$. The quantity $ E(\Sigma, X, T_0) $ itself depends on the choice of $X$, however the infimum of $ E(\Sigma, X, T_0)$ over $ T_0 $ does not. In particular, when $\Sigma$ lies in a time symmetric 3-slice in $N$ and has nonnegative Brown-York quasilocal mass $\mby(\Sigma)$, our estimates show that $\inf\limits_{T_0}E(\Sigma, X, T_0)$ equals $ \mby (\Sigma)$. We also study the spatial limit of $ \inf\limits_{T_0}E(S_r,X_r,T_0)$, where $S_r$ is a large coordinate sphere in a fixed end of an asymptotically flat initial data set $(M, g, p)$ and $X_r$ is an isometric embeddings of $S_r$ into $\mathbb{R}^3 \subset \mathbb{R}^{3,1}$. We show that if $(M, g, p)$ has future timelike ADM energy-momentum, then $\lim\limits_{r\to\infty}\inf\limits_{T_0}E(S_r,X_r,T_0)$ equals the ADM mass of $(M, g, p)$.