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:**


1. Does the Teichmuller metric and the hyperbolic metric for $g=1$ coincide?
2. If not, does it has nonpositive curvature?



What is the Teichmuller metric on the Teichmuller space of a closed surface of genus 1?
The answer is yes, up to renormalization.
Precisely, the bijection $\mathbb{H}^2\to Teich(\mathbb{T})$ you refer to induces an isometry from $(\mathbb{H}^2,d\_{\mathbb{H}^2})\to (Teich(\mathbb{T}),2d\_{Teich})$.
This is exactly Theorem 11.20 in Farb and Margalit's book A primer on mapping class groups, available [here](https://www.maths.ed.ac.uk/~v1ranick/papers/farbmarg.pdf).


