Ngô Quốc Anh

October 29, 2013

The mean curvature under conformal changes of Riemannian metrics: A global approach

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 22:46

In a previous post, I showed how the mean curvature changes under a conformal change using slightly local coordinates approach. Today, I want to reconsider that topic using global approach.

As usual, suppose (M,g) is an n-dimensional Riemannian manifold with boundary \partial M. We also assume that N is an outward unit normal vector field along the boundary \partial M. By an unit normal vector field we mean g(N,N)=1 and g(N,X)=0 for any tangent vectors X of \partial M. Note that \partial M is just a hypersurface of M and we also use g to denote the induced metric of g onto \partial M. Then we have the so-called second fundamental form \mathrm{I\!I} associated to \partial M defined to be

\displaystyle \mathrm{I\!I} (X,Y)=g(X,\nabla_Y N)

for any tangent vectors X,Y of \partial M.

TangentSpace

Regarding to the mean curvature H, we shall use the following definition

\displaystyle H=\text{trace}_g (\mathrm{I\!I})=g^{ij}\mathrm{I\!I}(\partial_i, \partial_j)=g^{ij} g(\partial_i,\nabla_{\partial_j} N).

Our aim is to calculate the mean curvature H under the following conformal change \widehat g=\phi^\kappa g for some smooth positive function \phi and a real number \kappa. It is important to note that by \widehat g=\phi^\kappa g we mean

\displaystyle \widehat g_{ij} = \phi^\kappa g_{ij}.

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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}}

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