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

1. Why do I have minus in the second equation (the second fundamental form changes under confromal metric)?

Comment by Han — June 8, 2018 @ 5:25

• Which equation?

Comment by Ngô Quốc Anh — June 8, 2018 @ 6:00

• The second one this page. When I write down the connection change under conformal metric, it ends up with ….-g_{ij}\nabla f.

Comment by Han — June 16, 2018 @ 1:36

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