Does moving a small enough distance in Teichmüller space change the marking?

Question

Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \rightarrow X$ is a quasiconformal homeorphism, quotient by a certain equivalence relation. We say that two such pairs $(X_{1},\phi_{1})$ and $(X_{2},\phi_{2})$ are equivalent if there exists a conformal map $h : X_{1} \rightarrow X_{2} $ such that $h \circ \phi_{1}$ is homotopic to $\phi_{2}$. We consider the Teichmüller distance $d_{T}$ defined on this space, where $d_{T} ([(X_{1},\phi_{1})],[(X_{2},\phi_{2})])=\frac{1}{2}logK_{h}$ such that $h : X_{1} \rightarrow X_{2}$ is the unique quasiconformal map in the homotopy class of $\phi_{2} \circ \phi_{1}^{-1}$ having the lowest quasiconstant $K_{h}$. Let us consider the point $x=[(S_{g}, id_{S_{g}})] \in \mathcal{T}(S_{g})$. Does there exist a positive real $r$ such that given any point $y=[(X,\phi)] \in B(x;r)$ there exists a pair $(Y,\psi)$ where $Y$ is a Riemann surface with underlying set same as $S_{g}$, $\psi : S_{g} \rightarrow Y$ is a quasiconformal homeomorphism homotopic to $id_{S_{g}}$ and $(X,\phi)$ and $(Y,\psi)$ are in the same equivalence class, i.e. $[(X,\phi)]=[(Y,\psi)]$ ?

Basically, what I am asking is that whether it's true that if we move a small enough distance from a point in Teichmüller space to some other point then the final point has the same making as the starting point.

Answer 1

Unless I misunderstand your question, you can take $r = \infty$. This is because we can choose representatives of marked conformal classes to be "Riemann surface structures" on $S_g$. Doing this, the marking maps are all the identity map.

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Basically, your equivalence classes are very very large. There is the Riemann surface which is the solutions to $x^4 + y^4 + z^4 = 0$ in $\mathbb{CP}^2$, and then there is that Riemann surface painted blue, and then there is that Riemann surface crossed with the first strongly inaccessible cardinal, and so on.

To shrink the equivalence classes a bit, we can instead think about Riemann surface structures (maximal atlases of charts) on a fixed surface. Now the marking map is clearer - it is the identity. However, the action of the mapping class group is harder to grasp. Also, our moduli space no longer contains "all" of the Riemann surfaces. Such is life.