Minimal surfaces + Semi-Geodesic Coordinates

Question

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

Answer 1

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 $\hat U\subset U$.