# 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 $\partial M$.

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

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

## 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

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