Ngô Quốc Anh

December 31, 2014

Conformal change of the Laplace-Beltrami operator

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:55

Happy New Year 2015!

In the last entry in 2014, I talk about conformal change of the Laplace-Beltrami operator. Given $(M,g)$ a Riemannian manifold of dimension $n \geqslant 2$. We denote $\widetilde g = e^{2\varphi} g$ a conformal metric of $g$ where the function $\varphi$ is smooth.

Recall the following formula for the Laplace-Beltrami operator $\Delta_g$ calculated with respect to the metric $g$:

$\displaystyle \Delta_g = \frac{1}{\sqrt{|\det g|}} \frac{\partial}{\partial x_j} \Big( \sqrt{|\det g|} g^{ij} \frac{\partial}{\partial x^i} \Big).$

where $\det g$ is the determinant of $g$. Then, it is natural to consider the relation between $\Delta_g$ and $\Delta_{\widetilde g}$ in terms of $\varphi$. Recall that by $\widetilde g = e^{2\varphi} g$ we mean, in local coordinates, the following

$\displaystyle \widetilde g_{ij} = e^{2\varphi} g_{ij},$

hence by taking the inverse, we obtain

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

Clearly,

$\displaystyle\det {\widetilde g} = e^{2n \varphi}\det g,$

hence

$\displaystyle\sqrt{| \det {\widetilde g} |} = e^{n \varphi} \sqrt{ |\det g| }.$

December 21, 2014

Conformal Changes of the Green function for the conformal Laplacian

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 11:00

Long time ago, I talked about conformal changes for various geometric quantities on a given Riemannian manifold $(M,g)$ of dimension $n$, see this post.

Frequently used in conformal geometry in general, or when solving the prescribed scalar curvature equation in particular, is the conformal Laplacian, defined as follows

$\displaystyle L_g(u) = - \frac{n-1}{4(n-2)}\Delta_g u + \text{Scal}_g u$

where $\text{Scal}_g$ is the scalar curvature of the metric $g$. The operator $L_g$ is conformal in the sense that any change of metric $\widehat g = \varphi ^\frac{4}{n-2}g$ would give the following magic identity

$\displaystyle L_{\widehat g} (u) =\varphi^{-\frac{n+2}{n-2}} L_g (\varphi u).$

Associated to the conformal Laplacian operator $L_g$ is the Green function, if exists, $\mathbb G_{L,g}$. Mathematically, the Green function $\mathbb G_{L,g}$ is defined to be a continuous function

$\mathbb G_{L,g} : M \times M \backslash \{(x,x) : x \in M\} \to \mathbb R$

such that for any $x\in M$, $\mathbb G_{L,g} (x, \cdot) \in L^1(M)$ and for any $u \in C^2(M)$ and any $x \in M$, we have the following representation

$\displaystyle u(x) = \int_M \mathbb G_{L,g}(x,y) L_g(u)(y) dv_g (y).$

December 4, 2014

Equations satisfied by standard bubbles and their derivatives in the Euclidean space

Filed under: Uncategorized — Ngô Quốc Anh @ 21:37

This note is purely involved calculation. In $\mathbb R^n$, let denote by $V_{(x,\varepsilon)} (y)$ the standard bubbles given by

$\displaystyle V_{(x,\varepsilon)} (y)= \left( \frac{\varepsilon}{\varepsilon^2+|y-x|^2}\right)^\frac{n-2}{2}.$

I am trying to derive some PDE for which the bubbles $V_{(x,\varepsilon)}$ solves.

1. First, we try to calculate $\Delta V_{(x,\varepsilon)}$. Clearly,

$\begin{array}{lcl} {\partial _{{y_i}}}{V_{(x,\varepsilon )}}(y) &=& \displaystyle \frac{{n - 2}}{2}{\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right)^{ - 1}}{V_{(x,\varepsilon )}}(y){\partial _{{y_i}}}\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right) \hfill \\ &=& \displaystyle -\frac{{n - 2}}{2}{\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right)^{ - 1}}{V_{(x,\varepsilon )}}(y)\frac{{2\varepsilon ({y_i} - {x_i})}}{{{{({\varepsilon ^2} + |y - x{|^2})}^2}}} \hfill \\ &=& \displaystyle -(n - 2)\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{V_{(x,\varepsilon )}}(y).\end{array}$

Taking derivative again gives

$\begin{array}{lcl} \partial _{{y_i}{y_i}}^2{V_{(x,\varepsilon )}}(y) &=& \displaystyle -(n - 2) {\partial _{{y_i}}}\left( {\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{V_{(x,\varepsilon )}}(y)} \right) \hfill \\ &=& \displaystyle -(n - 2) {V_{(x,\varepsilon )}}(y){\partial _{{y_i}}}\left( {\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}} \right) - (n - 2)\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{\partial _{{y_i}}}{V_{(x,\varepsilon )}}(y) \hfill \\ &=& \displaystyle -(n - 2) {I_1} - (n - 2) {I_2}.\end{array}$