In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies
$$ \int \nolimits _{0}^{t_*} (t_*-t) \Vert \omega (t)\Vert _{ L^\infty (B(x_{ *}, R_0))}\,{\hbox {d}}t <+\infty $$
for a ball
$$B(x_{ *}, R_0)$$
away from the axis of symmetry, then there exists no singularity at
$$t=t_*$$
in the torus
$$T(x_*, R)$$
generated by rotation of the ball
$$B(x_{ *}, R_0)$$
around the axis. This implies that possible singularity at
$$t=t_*$$
in the torus
$$T(x_*, R)$$
is excluded if the vorticity satisfies the blow-up rate
$$ \Vert \omega (t)\Vert _{L^\infty (T(x_*, R))}= O\left( \frac{1}{(t_*-t)^\gamma }\right) $$
as
$$t\rightarrow t_*$$
, where
$$\gamma <2$$
, and the torus
$$T(x_*, R)$$
does not touch the axis.