Related papers: Holomorphic contractibility of Teichm\"uller space…
The problem of holomorphic contractibilty of the Teichmuller spaces $T(0, n)$ of punctured spheres ($n > 4$) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author in…
We outline old and new results concerning the well-known problems in the Teichm\"uller space theory, i.e., whether these spaces are starlike in the Bers holomorphic embedding and whether any Teichm\"uller space of dimension greater than 1…
This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichm\"uller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for…
Based on the quasiconformal theory of the universal Teichm\"uller space, we introduce the Teichm\"uller space of diffeomorphisms of the unit circle with $\alpha$-H\"older continuous derivatives as a subspace of the universal Teichm\"uller…
In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $\pi_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichm\"uller…
We compute the factorisation homology of the four-punctured sphere and punctured torus over the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ explicitly as categories of equivariant modules using the framework of `Integrating Quantum…
The Teichm\"{u}ller curve is the fiber space over Teichm\"{u}ller space of closed Riemann surfaces, where the fiber over a point in Teichm\"{u}ller space is the underlying surface. We derive formulas for sectional curvatures on the…
In this paper, we discuss a rigidity property for holomorphic disks in Teichm\"uller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds.…
We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend…
For the twistor spaces of the Bochner-K\"ahler manifold $M = H^l \times P^n$, systems of holomorphic coordinates are constructed. As an application of them, an explicit description of the moduli space of relative deformations of fibers of…
In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that…
We address the existence of non-trivial closed invariant subspaces of operators $T$ on Banach spaces whenever their square $T^2$ have or, more generally, whether there exists a polynomial $p$ with $\mbox{deg}(p)\geq 2$ such that the lattice…
We study the class of holomorphic and isometric submersions between finite-type Teichm\"uller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map $\mathcal{T}_{g,n} \rightarrow…
We give an example of a horocycle in the Teichm\"uller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichm\"uller…
Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmuller spaces. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured…
We study the geometry of horospheres in Teichm\"uller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\geq 2$. We show that every $C^1$-diffeomorphism of Teichm\"uller space to itself that preserves horospheres is an…
It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a…
The problem of starlikeness of Teichmuller spaces in Bers' embedding was raised in 1974 and is solved (negatively) for Teichmuller spaces of sufficiently large dimensions. The original proof given by the author relies on the existence of…
We provide a complete classification of Teichm\"uller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichm\"uller curves intersect the boundary of the…
In this paper, the Teichm{\"u}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{\"u}ller spaces…