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Let $(X,\phi)$ be an element of Teichmuller space $\cal T_g$ and $q$ a (holomorphic) quadratic differential on $X$. Teichmuller geodesic flow gives a family of marked Riemann surfaces $(Y_t,\psi_t) = \exp(tq)$ for $t\in \mathbb R$. A quadratic differential $tq$ also determines a unique marked Riemann surface $(Y'_t,\psi'_t)$ and harmonic map $u_t:X\to Y_t'$ described below. Notice that when $t=0$, $X=Y_0=Y_0'$.

My question is: how $Y_t$ and $Y_t'$ are related. For example, is there some bound on the distance between them?

To describe how to get a Riemann surface from a quadratic differential, I am summarizing the discussion from Section 4.2 in Jurgen Jost's book Compact Riemann Surfaces:

Between any two points in Teichmuller space $(X,\phi),(Z,\psi)$ there is a unique harmonic map $u:X\to Z$ in the homotopy class of $\psi\circ \phi^{-1}$. Let $\rho^2dud\bar u$ be the hyperbolic metric on $Z$. Then $(\rho\circ u)^2 \frac{\partial u}{\partial z}\frac{\partial \bar u}{\partial z}dz^2$ is a holomorphic quadratic differential on $X$ and this map $\mathcal{ T}_g \to QD(X)$ between Teichmuller space and quadratic differentials is a bijection. Thus, from a quadratic differential $q$ on $X$, we get an element of Teichmuller space.

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