# Ngô Quốc Anh

## July 23, 2011

### The mean curvature under conformal changes of Riemannian metrics

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 7:58

Let $M$ be a Riemannian manifold of dimension $n$. On the boundary $\partial M$ we have the so-called outward normal vector $\eta$. Let $h_{ij}$ be the second fundamental form and

$\displaystyle h=\frac{1}{n-1}g^{ij}h_{ij}$

is the mean curvature. Let $\widetilde g = e^{2f}g$ be a metric conformally related to $g$. The transformation law for the second fundamental form reads as follows

$\displaystyle \widetilde h_{ij}=e^f h +\frac{\partial}{\partial\eta} (e^f)g_{ij}$

where $\frac{\partial}{\partial\eta}$ is the normal derivative with respect to $\eta$. Multiplying both sides of this equation with $\frac{1}{n-1}\widetilde g^{ij}$ gives

$\displaystyle \widetilde h=\frac{1}{n-1}\widetilde g^{ij}e^f h +\frac{1}{n-1}\widetilde g^{ij}\frac{\partial}{\partial\eta} (e^f)g_{ij},$

that is,

$\displaystyle \widetilde h = {e^{ - f}}h + {e^{ - f}}\frac{\partial }{{\partial \eta }}(f)$

since

$\displaystyle {\widetilde g^{ij}} = {e^{ - 2f}}{g^{ij}}$

and

$\displaystyle \frac{1}{{n - 1}}{g^{ij}}{g_{ij}} = 1.$

If we further let

$\displaystyle\widetilde g = {u^{\frac{4}{{n - 2}}}}g,$

that is, $f = \frac{2}{{n - 2}}\ln u$, we then have

$\displaystyle\widetilde h = {u^{ - \frac{2}{{n - 2}}}}\left( {h + \frac{\partial }{{\partial \eta }}(f)} \right) = \frac{2}{{n - 2}}\frac{{{u^{ - \frac{2}{{n - 2}}}}}}{u}\left( {\frac{{n - 2}}{2}hu + \frac{\partial }{{\partial \eta }}(u)} \right)$

since

$\displaystyle\frac{\partial }{{\partial \eta }}(f) = \frac{2}{{n - 2}}\frac{\partial }{{\partial \eta }}(\ln u) = \frac{2}{{n - 2}}\frac{{\frac{\partial }{{\partial \eta }}(u)}}{u}.$

In other words, we may write as

$\displaystyle\frac{{n - 2}}{2}hu + \frac{{\partial u}}{{\partial \eta }} = \frac{{n - 2}}{2}\widetilde h{u^{\frac{n}{{n - 2}}}}.$