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

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

July 15, 2014

Why should we call ” f=φg ” conformal change?

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:07

I asked this question to Professor Alice Chang when I met her during a conference in the University of Notre Dame this June. Loosely speaking, given a Riemannian manifold $(M,g)$, why frequently we follow the rule $f = \varphi g$ to change our metric in conformal geometry?

Professor Chang told me that it is because under the new metric, angles are preserved. The aim of this note is to make her answer clearer.

The best way to see this is to make use of the vector formulation of the law of cosines. Indeed, let us take two vectors $X$ and $Y$ sitting in the same tangent space, say $T_pM$ for some $p \in M$. Then the angel between these two vectors can be estimated as follows

$\displaystyle \cos_g(X,Y)=\frac{g(X,Y)}{\|X\|_g\|Y\|_g}$

where $\|\cdot\|_g$ is the norm evaluated with respect to the metric $g$. Under the new metric $f$ given by $\varphi g$, we first obtain

$\displaystyle \cos_f(X,Y)=\frac{f(X,Y)}{\|X\|_f\|Y\|_f}.$

Clearly, $f(X,Y) = \varphi g(X,Y)$. Moreover, if we write $X$ in local coordinates as $X=X^i \frac{\partial}{\partial x^i}$, we then have

$\displaystyle {\left\| X \right\|_f} = \sqrt {{{\left\langle {{X^i}\frac{\partial }{{\partial {x^i}}},{X^j}\frac{\partial }{{\partial {x^j}}}} \right\rangle }_f}} = \sqrt {{X^i}{X^j}} \sqrt {{f_{ij}}} = \sqrt \varphi \sqrt {{X^i}{X^j}} \sqrt {{g_{ij}}} .$

Therefore,

$\displaystyle {\left\| X \right\|_f}{\left\| Y \right\|_f} = \varphi {\left\| X \right\|_g}{\left\| Y \right\|_g},$

which immediately shows that $\cos_f(X,Y)=\cos_g(X,Y)$. In other words, the angle between the two vectors $X$ and $Y$ is preserved under the change of metrics.

May 16, 2010

Conformal Changes of Riemannian Metrics

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 5:54

I guess I will use the relation between curvature tensors of metrics lying in a conformal class frequently so I decide to post something related to this stuff which may be helpful and we can use later. Actually, I have used it when we proved conformal Laplacian operator is invariant. Let us briefly recall some terminologies

Definition (conformal). Two pseudo-Riemannian metrics $g$ and $\widetilde g$ on a manifold $M$ are said to be

1. (pointwise) conformal if there exists a $C^\infty$ function $f$ on $M$ such that

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

2. conformally equivalent if there exists a diffeomorphism $\alpha$ of $M$ such that $\alpha^* \widetilde g$ and $g$ are pointwise conformal.

Note that, if $g$ and $\widetilde g$ are conformally equivalent, then $\alpha$ is an isometry from $e^{2f}g$ onto $\widetilde g$. So we will only study below the case $\widetilde g = e^{2f}g$. Our aim is to compare Riemann curvature, Scalar curvature, Ricci curvature,… of $g$ and $\widetilde g$.

Definition (the Kulkarni–Nomizu product). This product $\odot$ is defined for two $(0,2)$-tensors and gives as a result a $(0,4)$-tensor. Precisely,

$\displaystyle \alpha \odot \beta (X_1,X_2,X_3,X_4)=\alpha (X_1,X_3)\beta (X_2,X_4)+\alpha (X_2,X_4)\beta (X_1,X_3)-\alpha (X_1,X_4)\beta (X_2,X_3)-\alpha (X_2,X_3)\beta (X_1,X_4)$

or

$\displaystyle {(\alpha \odot \beta )_{ijkl}} = {\alpha _{il}}{\beta _{jk}} + {\alpha _{jk}}{\beta _{il}} - {\alpha _{ik}}{\beta _{jl}} - {\alpha _{jl}}{\beta _{ik}}$.

Levi-Civita connection. On $(M,g)$, the Levi-Civita connection $\nabla$ is an affine connection which is torsion free

$\displaystyle \nabla_XY+\nabla_YX=[X,Y]$

and satisfies the rule

$\displaystyle X(g(Y,Z))=g(\nabla_X Y,Z) + g(Y, \nabla_X Z)$

for any vector fields $X,Y,Z$. We now have

$\displaystyle {\widetilde\nabla _X}Y = {\nabla _X}Y + X(f)Y + Y(f)X - g(X,Y) {\rm grad}f$.

Weyl tensor. This tensor is defined to be

$\displaystyle W = R - \frac{1}{{n - 2}}\left( {{\rm Ric} - \frac{{\rm Scal}}{n}g} \right) \odot g - \frac{{\rm Scal}}{{2n(n - 1)}}g \odot g$.

Thus we have the rule

$\displaystyle\widetilde W =W$.

Ricci tensor. This is a $(2,0)$-tensor defined by

$\displaystyle {\rm Ric}(X,Y) = {\rm Trace}( x \to R(x, X)Y)$.

In local coordinates, it has the form

$\displaystyle {\rm Ric} = R_{ij} dx^i \otimes dx^j$.

So we have the following rule

$\displaystyle\widetilde{\rm Ric} = {\rm Ric} - (n - 2)({\rm Hess} f -{\rm grad}f \otimes {\rm grad}f) + (\Delta f - (n - 2)|{\rm grad}f|^2)g$.

Traceless Ricci tensor. This tensor is defined by

$\displaystyle\displaystyle {Z_{ij}} = {R_{ij}} - \frac{1}{n}{\rm Scal}{g_{ij}}$.

A simple calculation shows that its trace, $g^{ij}Z_{ij}$, equals zero. So

$\displaystyle\widetilde Z = Z - (n - 2)\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f} \right) - \frac{{n - 2}}{n}\left( {\Delta f + |{\rm grad}f|^2} \right)g$.

Scalar curvature. This $(2,0)$ tensor is defined to be the trace of Ricci tensor, that is

$\displaystyle {\rm Scal} = {\rm Trace}( {\rm Ric}) = g^{jk}{\rm Ric}_{jk}$.

So

$\displaystyle\widetilde{\rm Scal} = {e^{ - 2f}}\left[ {{\rm Scal} + 2(n - 1)\Delta f - (n - 2)(n - 1)|{\rm grad} f{|^2}} \right]$.

In practice, this conformal change is not useful, we usually use the following conformal change

$\displaystyle \widetilde g=f^\frac{4}{n-2}g$.

With this, we simply have

$\displaystyle - \Delta f + \frac{{n - 2}}{4(n - 1)}{\rm Scal}f = \frac{{n - 2}}{4(n - 1)}\widetilde{{\rm Scal}}{f^{\frac{{n + 2}}{{n - 2}}}}$

or

$\displaystyle \widetilde{{\rm Scal}} = {f^{ - \frac{{n + 2}}{{n - 2}}}}\left[ { - \frac{4(n - 1)}{n - 2}\Delta f + {\rm Scal}f} \right]$.

Riemann curvature tensor. This $(1,3)$ tensor is defined to be

$\displaystyle R(X,Y)Z = \nabla_X \nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z$.

In local coordinates, we get

$\displaystyle R = R_{ikl}^j\dfrac{\partial }{{\partial {x^j}}} \otimes d{x^i} \otimes d{x^k} \otimes d{x^l}$.

So

$\displaystyle \widetilde R = {e^{2f}}\left[ {R - g\odot\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f + \frac{1}{2}|{\rm grad}f{|^2}g} \right)} \right]$.

Volume element. This, $d{\rm vol}_g$, is the unique density such that, for any orthonormal basis $(X_i)$ of $T_XM$,

$\displaystyle d{\rm vol}_g(X_1,...,X_n)=1$.

In local coordinates,

$\displaystyle d{\rm vol}_g = \sqrt{|g|} dx^1\wedge \dots \wedge dx^n$.

So

$\displaystyle d{\rm vol}_{\widetilde g} = e^{nf}d{\rm vol}_g$.

Hodge operator on $p$-forms (if $M$ is oriented). The Hodge star operator on an oriented inner product space $V$ is a linear operator on the exterior algebra of $V$, interchanging the subspaces of $k$-vectors and $n-k$-vectors where $n = \dim V$, for $0 \leqslant k \leqslant n$. It has the following property, which defines it completely: given an oriented orthonormal basis $e_1,e_2,\dots,e_n$ we have

$\displaystyle *(e_{i_1} \wedge e_{i_2}\wedge \cdots \wedge e_{i_k})= e_{i_{k+1}} \wedge e_{i_{k+2}} \wedge \cdots \wedge e_{i_n}$.

One can repeat the construction above for each cotangent space of an $n$-dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual $n-k$-form, of a $k$-form. The Hodge star then induces an $L^2$-norm inner product on the differential forms on the manifold. One writes

$\displaystyle (\eta,\zeta)=\int_M \eta\wedge *\zeta$

for the inner product of sections $\eta$ and $\zeta$ of $\Lambda^k(M)$. (The set of sections is frequently denoted as $\Omega^k(M) = \Gamma(\Lambda^k(M)$). Elements of $\Omega^k(M)$ are called exterior $k$-forms). For example, for a positively oriented orthogonal cofram $\{\omega^i\}_1^n$, one has

$\displaystyle *(\omega^1 \wedge \cdots \wedge \omega^p)=\omega^{p+1}\wedge \cdots \wedge \omega^n$.

So

$\displaystyle {*_{\widetilde g}} = {e^{(n - 2p)f}}{*_g}$.

Codifferential on $p$-forms. This notion $\delta$ is usually defined through the exterior derivative

$\displaystyle d : \Omega^p(M) \to \Omega^{p+1}(M)$

by the following rule (also called the formal adjoint of exterior derivative)

$\displaystyle\langle \eta,\delta \zeta\rangle = \langle d\eta,\zeta\rangle$,

i.e.

$\displaystyle \delta : \Omega^p(M) \to \Omega^{p-1}(M)$.

In other words, for a $p$-form $\beta$,

$\displaystyle \delta \beta = {( - 1)^{np + n + 1}}*d*\beta$.

So

$\displaystyle\widetilde\delta \beta = {e^{ - 2f}}\left[ {\delta \beta - (n - 2p){\iota_{{\rm grad}f}}\beta } \right]$

where $\iota$ denotes the interior product (the contraction of a differential form with a vector field).

(pseudo-) Laplacian on $p$-forms. This is known as the Hodge Laplacian and also known as the Laplace–de Rham operator. It is defined by

$\displaystyle\Delta= d\delta+\delta d$.

An important property of the Hodge Laplacian is that it commutes with the $*$ operator, i.e.

$\Delta * = * \Delta$.

So

$\displaystyle\widetilde\Delta \alpha = {e^{ - 2f}}\left[ {\Delta \alpha - (n - 2p)d({\iota_{{\rm grad}f}}\alpha ) - (n - 2p - 2){\iota_{{\rm grad}f}}d\alpha + 2(n - 2p){\rm grad}f \wedge {\iota_{{\rm grad}f}}\alpha - 2{\rm grad}f \wedge \delta \alpha } \right]$.

See also: Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1987.