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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

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  • $\begingroup$ there is a related result to this question which shows that it is always possible to build a conformal factor such that the geodesic spheres around a point will be of constant mean curvature locally... $\endgroup$ – Ali Nov 28 '14 at 23:32
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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 $\hat U\subset U$.

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  • $\begingroup$ Thanks Robert for your nice answer. May I ask you where the assumption about the positive injectivity radius was used? $\endgroup$ – Ali Nov 29 '14 at 21:03
  • $\begingroup$ Well, that probably was a bad use of terminology. All I meant to assume was that there is an $\epsilon>0$ such that the map $Z$ that you defined in your question is an injective diffeomorphism on $\Gamma\times (-\epsilon,\epsilon)\subset \Gamma\times\mathbb{R}$. That way, you can be sure that for $t$ sufficiently small, the level sets $\Gamma_t$ are disjoint. I didn't really want to assume anything about the injectivity radius of the induced metric on $\Gamma$. $\endgroup$ – Robert Bryant Nov 29 '14 at 21:21

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