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}}

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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}.

See also:

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

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