# What is the Teichmuller metric on the Teichmuller space of a closed surface of genus 1?

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?

• Is the metric well-defined for $g=1$? Dec 26, 2019 at 12:07
• @WillSawin Shouldn't it be? Are you saying it could vanish everywhere? Indeed I haven't seen any reference on this subject. Can you provide one or give a short explanation? Dec 26, 2019 at 12:10

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.