American Institute of Mathematical Sciences

Advanced
  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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Y. Canzani, On the multiplicity of the eigenvalues of the conformally covariant operators, E-print, arXiv:1207.0648, July 2012. Google Scholar

[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.  Google Scholar

[6]

Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants II: High-dimensional nullspace,, in preparation., ().   Google Scholar

[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.  Google Scholar

[8]

C. L. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.  Google Scholar

[13]

J. Kazdan and F. Warner, Scalar curvature and conformal deformations of Riemannian structure, J. Diff. Geom., 10 (1975), 113-134.  Google Scholar

[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.  Google Scholar

[15]

J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helvetici, 71 (1996), 213-228. doi: 10.1007/BF02566417.  Google Scholar

[16]

T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom., 25 (1987), 199-222.  Google Scholar

[17]

R. Ponge, Continuity and multiplicity of eigenvalues of Fredholm operators. Applications to conformally invariant operators,, in preparation., ().   Google Scholar

[18]

M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math., 52 (1999), 917-934.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Y. Canzani, On the multiplicity of the eigenvalues of the conformally covariant operators, E-print, arXiv:1207.0648, July 2012. Google Scholar

[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.  Google Scholar

[6]

Y. Canzani, A. R. Gover, D. Jakobson and R. Ponge, Conformal invariants II: High-dimensional nullspace,, in preparation., ().   Google Scholar

[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.  Google Scholar

[8]

C. L. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.  Google Scholar

[13]

J. Kazdan and F. Warner, Scalar curvature and conformal deformations of Riemannian structure, J. Diff. Geom., 10 (1975), 113-134.  Google Scholar

[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.  Google Scholar

[15]

J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helvetici, 71 (1996), 213-228. doi: 10.1007/BF02566417.  Google Scholar

[16]

T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom., 25 (1987), 199-222.  Google Scholar

[17]

R. Ponge, Continuity and multiplicity of eigenvalues of Fredholm operators. Applications to conformally invariant operators,, in preparation., ().   Google Scholar

[18]

M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math., 52 (1999), 917-934.  Google Scholar

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