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| }.

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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).

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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}

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