# Conformally invariant operators

(Difference between revisions)
 Revision as of 20:32, 23 September 2013 (view source)Tianling (Talk | contribs)← Older edit Revision as of 21:08, 23 September 2013 (view source)Tianling (Talk | contribs) Newer edit → Line 27: Line 27: 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 : 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 for $k>\frac n2$ and $n\ge 4$ even. + * GJMS operators : 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 for $k>\frac n2$ and $n\ge 4$ even. + *Scattering operators , or the conformally invariant fractional powers of the Laplacian : 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 reconciled the way of defining $P_\sigma$ in and the localization method of Caffarelli-Silvestre  for the fractional Laplacian in the Euclidean space $\mathbb{R}^n$. + + + Special cases: + + * On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians. + + * On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator of explicit formula: + $+ P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2}, +$ + 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, + + * has the eigenfunctions of spherical harmonics, and + + * is the inverse of a spherical Riesz potential. Line 34: Line 51: {{reflist|refs= {{reflist|refs= - {{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}} + {{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}} + + {{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}} + + {{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}} + + {{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}} {{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}} {{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}} + + {{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}} {{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}} {{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}}

## Revision as of 21:08, 23 September 2013

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.

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 [1] [2]:

$P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,$ 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 [3]: 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 [4] for $k>\frac n2$ and $n\ge 4$ even.
• Scattering operators [5], or the conformally invariant fractional powers of the Laplacian [6]: 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 [6] reconciled the way of defining $P_\sigma$ in [5] and the localization method of Caffarelli-Silvestre [7] for the fractional Laplacian in the Euclidean space $\mathbb{R}^n$.

Special cases:

• On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
• On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator [8] of explicit formula:

$P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},$ 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,

* has the eigenfunctions of spherical harmonics, and

* is the inverse of a spherical Riesz potential.


## References

1. Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint
2. Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4)
3. Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence", Journal of the London Mathematical Society 2: 557--565
4. Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem", Journal of the American Mathematical Society 17: 389--405
5. 5.0 5.1 Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry", Inventiones mathematicae 152: 89--118
6. 6.0 6.1 Chang, Sun-Yung Alice; González, Mar\'\ia del Mar (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226: 1410--1432
7. Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32: 1245--1260
8. Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry", Journal of functional analysis 74: 199--291