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In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the Nielsen problem" here, Wolpert gave an alternative proof which includes the convexity of length functions for surfaces with punctures.

Kerckhoff's proof shows convexity along earthquake paths, on the other hand Wolpert proved the convexity along Weil-Petersson Geodesics. So the question is

1) Is the length function convex along earthquake paths on the Teichmüller spaces of surfaces with boundary and punctures?

2) Is the length function convex along Weil-Petersson geodesics on the Teichmüller spaces of surfaces with boundary?

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  • $\begingroup$ These two results deal with two different notions of convexity for functions. Which one you are actually interest in? $\endgroup$
    – Misha
    Commented Jul 31, 2016 at 16:48
  • $\begingroup$ @Misha Thanks for pointing out the confusion. I have edited the question. $\endgroup$
    – Cusp
    Commented Aug 1, 2016 at 6:12

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