Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through the local diffeomorhism $Z:\Gamma \times \mathbb{R} \to M$ 
$ Z(y,z) = Exp\_y (z N(y)) $ where $N$ is the unit vector along $\Sigma$


Define $\Gamma\_{t} = \left\{ {x\_3=t} \right\} $. and let $H\_{x\_3}$ denote the mean curvature of $\Gamma\_{x\_3}$


Does there exist a conformal factor $c$ such that if $ \hat{g} = c g$ then $\hat{H\_t}$ is constant for all sufficiently small t?


Thanks



Minimal surfaces + Semi-Geodesic Coordinates
Yes, this can always be done as long as $\Gamma$ has a positive injectivity radius. 


In fact, calculation yields the following formula: Let $V$ be the unit vector field (defined in a neighborhood of $\Gamma$) that is $g$-perpendicular to the level sets of $t$ (i.e., $V$ is perpendicular to each $\Gamma\_t$).
This $V$ is well-defined on a $\Gamma$-neighborhood $U\subset M$. Then 
$$
\mathrm{d}c(V) = 2 H\,c - 2\hat H\,c^{3/2},
$$
where $H$ is the function on $U$ that gives the mean curvature of the level sets $\Gamma\_t$ in the metric $g$ and $\hat H$ is the function on $U$ that gives the mean curvature of the level sets $\Gamma\_t$ in the metric $\hat g = c g$. 


Thus, one can specify any desired function $\hat H$ (in particular, a function constant on the each level set $\Gamma\_t$) and choose any desired initial function $c\_0>0$ on $\Gamma=\Gamma\_0$, and there will be a unique solution $c>0$ to the above equation (which is a first order PDE) that satisfies $c = c\_0$ along $\Gamma\_0$. This solution $c$ will be defined on an open neighborhood $\hat U\subset U$.


