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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?

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    $\begingroup$ Is the metric well-defined for $g=1$? $\endgroup$
    – Will Sawin
    Dec 26, 2019 at 12:07
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    $\begingroup$ @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? $\endgroup$
    – trisct
    Dec 26, 2019 at 12:10

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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.

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