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