Howard Masur's research asserts that if $S_g$ is a closed surface of genus $g\geq2$, then the Teichmuller space $T(S_g)$ does not have nonpositive curvature. His proof relies on the existence of similar Strebel rays. However, similar Strebel rays does not exist in $T(S_g)$ if $g=1$. On the other hand, $T(S_1)=\mathbb H$ does admit a hyperbolic metric.

**Questions:**

Does the Teichmuller metric and the hyperbolic metric for $g=1$ coincide?

If not, does it has nonpositive curvature?