Weak Minimal Area In Entanglement Entropy

We re-visit the minimal area condition of Ryu-Takayanagi in the holographic calculation of the entanglement entropy. In particular, the Legendre test and the Jacobi test. The necessary condition for the weak minimality is checked via Legendre test and its sufficient nature via Jacobi test. We show for AdS black hole with a strip type entangling region that it is this minimality condition that makes the hypersurface not to cross the horizon, which is in agreement with that studied earlier by {\it Engelhardt et al.} and {\it Hubeny} using a different approach. Moreover, demanding the weak minimality condition on the entanglement entropy functional with the higher derivative term puts a constraint on the Gauss-Bonnet coupling: that is there should be an upper bound on the value of the coupling, $\lambda_a< \f{(d-3)}{4(d-1)}$.