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It is known that Teichmüller distance ($d_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d_{WP}$) is not complete.

See for example the article

Wolpert, Scott. Noncompleteness of the Weil-Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), no. 2, 573--577 ,

in which Wolpert constructs an the explicit path on the Teichmüller space, which diverges in $d_{Teich}$, but converges in $d_{WP}$.

My question is : is it true in general that there is a positive constant $C$ such that $d_{WP} < C d_{Teich}$?

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Yes, this is a result of Michele Linch, 1974.

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