# 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

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

## July 15, 2011

### Stereographic projection, 4

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 19:31

It turns out that, via the stereographic projection, equation

$\displaystyle - {\Delta _{{\mathbb S^n}}}u = \lambda u + {u^{\frac{{n + 2}}{{n - 2}}}}$

on $\mathbb S^n$ with $u>0$ becomes

$\displaystyle - {\Delta _{{\mathbb R^n}}}u = V(x) u + {u^{\frac{{n + 2}}{{n - 2}}}}, \quad x \in \mathbb R^n$

with the following property

$u(x) \to 0, \quad |x|\to +\infty$

where

$\displaystyle V(x) = \frac{{n(n - 2) + 4\lambda }}{{{{(1 + |x|^2)}^2}}}.$

A very simple consequence is that for the prescribing scalar curvature equation, the term $V$ disappears as we already notice that

$\displaystyle\lambda = -\frac{n(n-2)}{4}.$

## July 11, 2011

### Energy functionals associate to integrals of exponential type

Filed under: PDEs — Ngô Quốc Anh @ 15:05

The purpose of this note is to derive some integral functionals $\mathcal F$ associated to the following

$\displaystyle R(x)e^u$

in the weak form in the sense that each critical point of $\mathcal F$ is a weak solution for equation

$\displaystyle R(x)e^u = 0.$

For simplicitly, we denote

$\displaystyle \mathcal I(u) = \int_M R(x)e^udv_g$

where $M$ is a Riemannian manifold with metric $g$ and $u$ a function sitting in an appropriate Sobolev space. To be exact, we shall find a functional $\mathcal F$ so that its first variation, denoted by $\delta\mathcal F$, equals $\mathcal I$.

Type 1. We shall find $\mathcal F$ of the following form

$\displaystyle \mathcal F(u) = C\int_M R(x)e^udv_g$