Maximal open radius for Strebel point
Given a marked surface \([f]\) in the Teichmüller space \(\mathcal{T}(S)\) for a Riemann surface \(S\) (with negative Euler characteristic), recall the extremal maximal dilatation \[ K_0([f]):=\inf_{g\in[f]} \left\{\frac{1+\|\mu_g\|}{1-\|\mu_g\|}\right\} \] and the boundary dilatation \[ H([f]):=\inf_{g\in[f]} \inf_{E\subset_c S} \left\{\frac{1+\|\mu_g|_{S-E}\|}{1-\|\mu_g|_{S-E}\|}\right\}\leq K_0([f]), \] where the inner infimum is taken over the collection of compact subsets \(E\) in \(S\). A marked surface \([g]\) is called a Strebel point if its extremal maximal dilation is strictly greater than its boundary dilatation, and non-Strebel point if the two are equal. This is a closed condition, and the complement set \(SP\) of Strebel points in any infinite-dimensional Teichmüller space \(\mathcal{T}(S)\) is open and dense. In this paper, the author helps to elucidate the geometry of \(SP\) by expressing the radius \(R([f])\), with respect to the Teichmüller distance, of the largest open ball centered at an arbitrary Strebel point \([f]\in\mathcal{T}(S)\) in terms of its extremal maximal dilatation and its boundary dilatation: \[ R([f])=\frac{1}{4}\log\left(\frac{K_0([f])}{H([f])}\right). \] Direct computation shows that everything in the open ball of radius \(R([f])\) is contained in \(SP\), and the author constructs non-Strebel points arbitrarily close to the sphere based at \([f]\) with radius \(R([f])\), showing that this is indeed the maximal radius. This is necessary (as opposed to constructing a non-Strebel point on the boundary) because he further proves that the sphere about \([f]\) of radius \(R([f])\) is contained in \(SP\). This is a striking result, and topologically possible because \(\mathcal{T}(S)\) is infinite-dimensional. The proof involves showing that, having a non-Strebel point \([g]\) on the radius \(R([f])\) sphere necessarily results in a geodesic, composed of non-Strebel points joining \([f]\) and \([g]\). The author also provides infinitesimal versions of his results for Beltrami differentials. The author's writing is succinct and clear, although there is a minor error (lines 17 and 18 on page 323) to the proof of his infinitesimal result. He has offered up the following correction: \[ \begin{aligned} (t-1)|\mu(z)|&\leq (t-1)(b^*(\mu)+\epsilon')\\ &\leq \frac{b([\mu]_B)+\epsilon'}{b([\mu]_B)}\left(\frac{\|\mu\|+b([\mu]_B)+\epsilon'}{2}-b([\mu]_B)\right) \leq \delta +C\epsilon' \end{aligned} \] for \(z\in S-E\), where \(C\) is a suitable constant depending on \(b([\mu]_B)\) and \(\|\mu\|\).
scientific article; zbMATH DE number 6488311