Conformally invariant operators - Revision history

imported>Tianling at 05:04, 24 September 2013

2013-09-24T05:04:32Z

← Older revision		Revision as of 00:04, 24 September 2013
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	* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.		* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.

	*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. ~~The authors~~ <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.		*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. Chang and González <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in Graham-Zworski <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

imported>Tianling at 03:05, 24 September 2013

2013-09-24T03:05:58Z

← Older revision		Revision as of 22:05, 23 September 2013
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	\]		\]
	where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$		where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$
	* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,		* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,

	* has the eigenfunctions of spherical harmonics, and		* has the eigenfunctions of spherical harmonics, and

	* is the inverse of a spherical Riesz potential.		* is the inverse of a spherical Riesz potential.

imported>Tianling at 03:05, 24 September 2013

2013-09-24T03:05:04Z

← Older revision		Revision as of 22:05, 23 September 2013
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	This is a fourth order operator with leading term $(-\Delta_g)^2$.		This is a fourth order operator with leading term $(-\Delta_g)^2$.

	* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even~~. The formula are not explicit except they are on the standard sphere~~. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.		* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.

	*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.		*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.
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	<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= Maria del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>		<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= Maria del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>

			<ref name="FG13">{{Citation \| last1=Fefferman \| first1= Charles \| last2=Graham \| first2= C \| title=Juhl’s formulae for GJMS operators and 𝑄-curvatures \| journal=Journal of the American Mathematical Society\|year=2013 \| volume=26 \| pages=1191--1207}}</ref>

	<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>		<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>
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	<ref name="graham2003scattering">{{Citation \| last1=Graham \| first1= C Robin \| last2=Zworski \| first2= Maciej \| title=Scattering matrix in conformal geometry \| journal=Inventiones mathematicae \| year=2003 \| volume=152 \| pages=89--118}}</ref>		<ref name="graham2003scattering">{{Citation \| last1=Graham \| first1= C Robin \| last2=Zworski \| first2= Maciej \| title=Scattering matrix in conformal geometry \| journal=Inventiones mathematicae \| year=2003 \| volume=152 \| pages=89--118}}</ref>

			<ref name="Juhl1">{{Citation \| last1=Juhl \| first1= Andreas \| title=On the recursive structure of Branson’s Q-curvature \| journal=arXiv preprint arXiv:1004.1784}}</ref>

			<ref name="Juhl2">{{Citation \| last1=Juhl \| first1= Andreas \| title=Explicit formulas for GJMS-operators and Q-curvatures \| journal=Geometric and Functional Analysis \| year=2013\|volume=23 \| pages=1278--1370}}</ref>

	<ref name="paneitz1983quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds \|year=1983 \| journal=preprint}}</ref>		<ref name="paneitz1983quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds \|year=1983 \| journal=preprint}}</ref>

imported>Tianling at 02:55, 24 September 2013

2013-09-24T02:55:08Z

← Older revision		Revision as of 21:55, 23 September 2013
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	On a general compact manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by		On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by
	\[		\[
	A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),		A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
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	This is a fourth order operator with leading term $(-\Delta_g)^2$.		This is a fourth order operator with leading term $(-\Delta_g)^2$.

	* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.		* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.

	*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.		*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

imported>Tianling at 21:51, 23 September 2013

2013-09-23T21:51:32Z

← Older revision		Revision as of 16:51, 23 September 2013
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	<ref name="CSextension">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=An extension problem related to the fractional Laplacian \| journal=Communications in Partial Differential Equations \| year=2007 \| volume=32 \| pages=1245--1260}}</ref>		<ref name="CSextension">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=An extension problem related to the fractional Laplacian \| journal=Communications in Partial Differential Equations \| year=2007 \| volume=32 \| pages=1245--1260}}</ref>

	<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= ~~Mar\'\ia~~ del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>		<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= Maria del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>

	<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>		<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>

imported>Tianling at 21:49, 23 September 2013

2013-09-23T21:49:44Z

← Older revision		Revision as of 16:49, 23 September 2013
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	* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.		* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.

	*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian in the Euclidean space $\mathbb{R}^n$.		*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

imported>Tianling at 21:08, 23 September 2013

2013-09-23T21:08:27Z

@@ Line 27: / Line 27: @@
 This is a fourth order operator with leading term $(-\Delta_g)^2$.
-* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.
+* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. The formula are not explicit except they are on the standard sphere. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.
@@ Line 34: / Line 51: @@
 {{reflist|refs=
-<ref name="gover2004conformally">{{Citation | last1=Gover | first1= A | last2=Hirachi | first2= Kengo | title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem | journal=Journal of the American Mathematical Society | volume=17 | pages=389--405}}</ref>
+<ref name="branson1987group">{{Citation | last1=Branson | first1= Thomas P | title=Group representations arising from Lorentz conformal geometry | journal=Journal of functional analysis | year=1987 | volume=74 | pages=199--291}}</ref>
 <ref name="GJMS">{{Citation | last1=Graham | first1= C Robin | last2=Jenne | first2= Ralph | last3=Mason | first3= Lionel J | last4=Sparling | first4= George AJ | title=Conformally invariant powers of the Laplacian, I: Existence | journal=Journal of the London Mathematical Society | year=1992 | volume=2 | pages=557--565}}</ref>
 <ref name="paneitz1983quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}}</ref>

imported>Tianling at 20:32, 23 September 2013

2013-09-23T20:32:31Z

← Older revision		Revision as of 15:32, 23 September 2013
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	Examples of conformally invariant operators include:		Examples of conformally invariant operators include:

	* The conformal Laplacian: $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.		* The conformal Laplacian:
			\[
			L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,
			\]
			where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

	* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>.		* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
			\[
			P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
			\]
			where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor,
			\[
			Q=c_n\|Ric_g\|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR
			\]
			and
			\[
			a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}.
			\]
			This is a fourth order operator with leading term $(-\Delta_g)^2$.

			* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ is $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even.



			== References ==
			{{reflist\|refs=

			<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \| volume=17 \| pages=389--405}}</ref>

			<ref name="GJMS">{{Citation \| last1=Graham \| first1= C Robin \| last2=Jenne \| first2= Ralph \| last3=Mason \| first3= Lionel J \| last4=Sparling \| first4= George AJ \| title=Conformally invariant powers of the Laplacian, I: Existence \| journal=Journal of the London Mathematical Society \| year=1992 \| volume=2 \| pages=557--565}}</ref>

	~~== References ==~~
	~~{{reflist\|refs=~~
	<ref name="paneitz1983quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds \|year=1983 \| journal=preprint}}</ref>		<ref name="paneitz1983quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds \|year=1983 \| journal=preprint}}</ref>

imported>Tianling at 19:48, 23 September 2013

2013-09-23T19:48:27Z

← Older revision		Revision as of 14:48, 23 September 2013
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	\]		\]
	where $a, b$ are constant.		where $a, b$ are constant.

			Examples of conformally invariant operators include:

			* The conformal Laplacian: $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

			* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>.






			== References ==
			{{reflist\|refs=
			<ref name="paneitz1983quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds \|year=1983 \| journal=preprint}}</ref>

			<ref name="paneitz2008quartic">{{Citation \| last1=Paneitz \| first1= S \| title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) \| url=http://dx.doi.org/10.3842/SIGMA.2008.036 \| doi:10.3842/SIGMA.2008.036 \| year=2008 \| journal=SIGMA Symmetry Integrability Geom. Methods Appl. \| issue=4 \| Paper=036}}</ref>

			}}

imported>Tianling: Created page with "On a general compact manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$..."

2013-09-23T19:15:23Z

Created page with "On a general compact manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$..."

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On a general compact manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by
\[
A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
\]
where $a, b$ are constant.

← Older revision		Revision as of 16:51, 23 September 2013
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	<ref name="CSextension">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=An extension problem related to the fractional Laplacian \| journal=Communications in Partial Differential Equations \| year=2007 \| volume=32 \| pages=1245--1260}}</ref>		<ref name="CSextension">{{Citation \| last1=Caffarelli \| first1= Luis \| last2=Silvestre \| first2= Luis \| title=An extension problem related to the fractional Laplacian \| journal=Communications in Partial Differential Equations \| year=2007 \| volume=32 \| pages=1245--1260}}</ref>

	<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= ~~Mar\'\ia~~ del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>		<ref name="chang2011fractional">{{Citation \| last1=Chang \| first1= Sun-Yung Alice \| last2=González \| first2= Maria del Mar \| title=Fractional Laplacian in conformal geometry \| journal=Advances in Mathematics \| year=2011 \| volume=226 \| pages=1410--1432}}</ref>

	<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>		<ref name="gover2004conformally">{{Citation \| last1=Gover \| first1= A \| last2=Hirachi \| first2= Kengo \| title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem \| journal=Journal of the American Mathematical Society \|year=2004 \|volume=17 \| pages=389--405}}</ref>