Nullspaces of conformally invariant operators. Applications to \boldsymbol{Q_k}-curvature

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2013, 20: 43-50. doi: 10.3934/era.2013.20.43

Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature

Yaiza Canzani ^1, , A. Rod Gover ^2, , Dmitry Jakobson ^1, and Raphaël Ponge ^3,

1.	Department of Mathematics and Statistics, McGill University, Montréal, Canada, Canada
2.	Department of Mathematics, University of Auckland, New Zealand &, Mathematical Sciences Institute, Australian National University, Canberra, Australia
3.	Department of Mathematical Sciences, Seoul National University, Seoul, South Korea

Received June 2012 Revised March 2013 Published March 2013

Abstract
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We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.

Keywords: conformal geometry, Spectral geometry, Qk-curvature., nodal sets.

Mathematics Subject Classification: 58J50, 53A30, 53A55, 53C2.

Citation: Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43

References:

[1]	P. Baird, A. Fardoun and R. Regbaoui, Prescribed Q-curvature on manifolds of even dimension, J. Geom. Phys., 59 (2009), 221-233. doi: 10.1016/j.geomphys.2008.10.007.
[2]	T. Branson and B. Ørsted, Conformal geometry and global invariants, Differential Geometry and its Applications, 1 (1991), 279-308. doi: 10.1016/0926-2245(91)90004-S.
[3]	S. Brendle, Convergence of the $Q$-curvature flow on $S^4$, Adv. Math., 205 (2006), 1-32. doi: 10.1016/j.aim.2005.07.002.
[4]	Y. Canzani, On the multiplicity of the eigenvalues of the conformally covariant operators, E-print, arXiv:1207.0648, July 2012.
[5]	Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants from nodal sets. I. Negative Eigenvalues and Curvature Prescription, Int. Mat. Res. Notices, (2013). doi: 10.1093/imrn/rns295.
[6]	Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants II: High-dimensional nullspace,, in preparation., ().
[7]	Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant $Q$-curvature, Ann. of Math. (2), 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813.
[8]	C. L. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.
[9]	A. R. Gover, Q curvature prescription; forbidden functions and the GJMS null space, Proc. Amer. Math. Soc., 138 (2010), 1453-1459. doi: 10.1090/S0002-9939-09-10111-9.
[10]	C. S. Gordon and E. N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. doi: 10.1307/mmj/1029003354.
[11]	C. R. Graham, R. Jenne, L. J. Mason and G. A. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc. (2), 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.
[12]	C. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.
[13]	J. Kazdan and F. Warner, Scalar curvature and conformal deformations of Riemannian structure, J. Diff. Geom., 10 (1975), 113-134.
[14]	K. Kodaira and D. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43-76. doi: 10.2307/1969879.
[15]	J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helvetici, 71 (1996), 213-228. doi: 10.1007/BF02566417.
[16]	T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom., 25 (1987), 199-222.
[17]	R. Ponge, Continuity and multiplicity of eigenvalues of Fredholm operators. Applications to conformally invariant operators,, in preparation., ().
[18]	M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math., 52 (1999), 917-934.

show all references

References:

[1]	P. Baird, A. Fardoun and R. Regbaoui, Prescribed Q-curvature on manifolds of even dimension, J. Geom. Phys., 59 (2009), 221-233. doi: 10.1016/j.geomphys.2008.10.007.
[2]	T. Branson and B. Ørsted, Conformal geometry and global invariants, Differential Geometry and its Applications, 1 (1991), 279-308. doi: 10.1016/0926-2245(91)90004-S.
[3]	S. Brendle, Convergence of the $Q$-curvature flow on $S^4$, Adv. Math., 205 (2006), 1-32. doi: 10.1016/j.aim.2005.07.002.
[4]	Y. Canzani, On the multiplicity of the eigenvalues of the conformally covariant operators, E-print, arXiv:1207.0648, July 2012.
[5]	Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants from nodal sets. I. Negative Eigenvalues and Curvature Prescription, Int. Mat. Res. Notices, (2013). doi: 10.1093/imrn/rns295.
[6]	Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants II: High-dimensional nullspace,, in preparation., ().
[7]	Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant $Q$-curvature, Ann. of Math. (2), 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813.
[8]	C. L. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.
[9]	A. R. Gover, Q curvature prescription; forbidden functions and the GJMS null space, Proc. Amer. Math. Soc., 138 (2010), 1453-1459. doi: 10.1090/S0002-9939-09-10111-9.
[10]	C. S. Gordon and E. N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. doi: 10.1307/mmj/1029003354.
[11]	C. R. Graham, R. Jenne, L. J. Mason and G. A. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc. (2), 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.
[12]	C. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.
[13]	J. Kazdan and F. Warner, Scalar curvature and conformal deformations of Riemannian structure, J. Diff. Geom., 10 (1975), 113-134.
[14]	K. Kodaira and D. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43-76. doi: 10.2307/1969879.
[15]	J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helvetici, 71 (1996), 213-228. doi: 10.1007/BF02566417.
[16]	T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom., 25 (1987), 199-222.
[17]	R. Ponge, Continuity and multiplicity of eigenvalues of Fredholm operators. Applications to conformally invariant operators,, in preparation., ().
[18]	M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math., 52 (1999), 917-934.

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