Let $f:(S,g) \to (M,h) $ be a smooth immersion of a compact surface into a 3 - manifold. Is it true that there exists a diffeomorphism $\phi: S \to S$, such that the metric $(f \circ \phi)^*(h)$ is conformal to $g$ ?
To explain the motivation behind ths question, consider first the following set-up. Let $D_2 \subset \mathbb R^2$ be the standard 2-disk and $f: (S,g_{eukl}) \to (M,g)$ an immersion into a Riemannian manifold. It is well-known that the area of $f$ is defined as $A(f) = \int_{D_2} \sqrt{||f_1||_g \cdot ||f_2||_g - g(f_1,f_2)^2 } dx_1dx_2$, where $f_i := \partial f/\partial x_i$. On the other hand, the Dirichlet energy is defined as $E(f) = \frac{1}{2}\int_{D_2} ||f_1||_g + ||f_2||_g dx_1dx_2$, and by simply comparing the integrants, one sees that $A(f) \leq E(f)$ with equality holding if and only if $||f_1||_g = ||f_2||_g$ and $g(f_1,f_2) = 0$ everywhere. Immersions with that property are called conformal and it is well known that for any given immersion, there exists a parameter transformation $\psi:D_2 \to D_2$ such that $f \circ \psi$ is a conformal immersion. In other words, an immersion $f$ is conformal if and only if $f^*(g) = e^{\phi} \cdot g_{eukl}$ for some smooth function $\phi$ on $D_2$, which means precisely that $f^*(g)$ and $g_{eukl}$ are conformal as metrics on $D_2$.
I have some trouble understanding why this is still possible when $f:(S,h) \to (M,g)$ is an immersion of any compact Riemannian 2-manifold. I know that the area of $f$ is in general defined by $A(g) := \int_S \omega_{f^*(g)}$, whereas the Dirichlet energy depends on $h$ and is defined by $E_h(f) := \frac{1}{2} \int_S |df|^2 \omega_h$. I have shown that we always have $A(f) \leq E_h(f)$ (regardless of $h$), where $f$ is said to be conformal if equality holds. One can show that this is equivalent to the metrics $f^*(g)$ and $h$ being conformal, i.e $f^*(g) = e^\phi \cdot h$. Is it still true that there exists a diffeomorphism $\psi: S \to S$ such that $f \circ \psi$ is conformal ? I ask because I only find papers that assert the existence of isothermal coordinates on any Riemannian 2-manifold, which essentially seems to only imply the case from the first paragraph.
You are asking whether there is a diffeomorphism $ \psi : S\to S$ so that $f\circ \psi$ is conformal. Writing $g_1 = f^* h$, it's the same as asking whether $ \psi^* g _1$ is conformal to $g$.
Note that $g$, $g_1$ induces two conformal structures on $S$. In general they are not conformal to each other. Indeed, the dimension of the moduli space of conformal structures on a surface of genus $g\ge 2$ is $6g-6$, and is $2$ when $g= 1$.
The only case you can find such a $\psi$ (for a compact surface $S$) is when $S$ is topologically a 2-sphere. This is given by the uniformization theorem, which says that $S$ is conformal to the standard $\mathbb S^2$ in $\mathbb R^3$.
Note that the theorem on the existence of isothermal coordinates can be interpreted as saying that there is only one conformal structure on the unit disk.