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}}}}.

See also:

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