By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is never of the class of $C^2$-smooth. What about the lower regularity? For instance, do all Teichmüller domains have Lipschitz boundary?
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asked Mar 16 at 15:18
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2$\begingroup$ Conjecurally, the boundary is not even locally connected. $\endgroup$– Moishe KohanCommented Mar 16 at 15:50
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1$\begingroup$ @Mahdi - This is not an answer to your question but perhaps is moving in the correct direction: the boundary of the Bers embedding is not $C^1$. See the paper linked to here: mathoverflow.net/questions/464087/… $\endgroup$– Sam NeadCommented Mar 16 at 16:41
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2$\begingroup$ @SamNead: Ken Bromberg's paper where he proves that the boundary of the quasifuchsian space is not locally connected. $\endgroup$– Moishe KohanCommented Mar 16 at 16:48
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1$\begingroup$ @SamNeed: A paper by Subhojoy Gupta and Harish Seshadri entitled "On domains biholomorphic to Teichmüller spaces" has many interesting things. arxiv.org/abs/1701.06860 See also the reference [Yau11] in their paper. $\endgroup$– Mahdi Teymuri GarakaniCommented Mar 16 at 16:55
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2$\begingroup$ @SamNead: Bromberg's paper is in Duke Math Journal, 2011. $\endgroup$– Moishe KohanCommented Mar 16 at 17:15
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1 Answer
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To follow up on Moishe Kohan's comment (and my own):
Bromberg conjectures, in his 2011 paper The space of kleinian punctured torus groups is not locally connected, that the same holds for any surface $S$ (with negative Euler characteristic). This is proved by Magid in his 2012 paper Deformation spaces of kleinian surface groups are not locally connected.
More directly related to your question is Bromberg's other conjecture (in the same paper): that the boundary of a Bers slice is not locally connected (when $S$ has Euler characteristic less than minus one). This appears to still be open.
I'll end by remarking that Goodman shows, in his 2006 paper, Spirals in the boundary of slices of quasi-fuchsian space that the boundary of the Bers slice is not $C^1$.
