Rigidity of conformal functionals on spheres

TY - JOUR

T1 - Rigidity of conformal functionals on spheres

AU - Møller, Niels Martin

AU - Ørsted, Bent

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In this paper, we investigate the stationary points of functionals on the space of Riemannian metrics on a compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric and the manifold is the n-sphere, we apply methods from representation theory to give a universal form of the Hessian at a stationary point. This reveals a very strong rigidity in the local structure of any such functional. As a corollary, this gives a new proof of the results of Okikiolu on local maxima and minima for the determinant of the conformal Laplacian, and we obtain results of the same type in general examples.

AB - In this paper, we investigate the stationary points of functionals on the space of Riemannian metrics on a compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric and the manifold is the n-sphere, we apply methods from representation theory to give a universal form of the Hessian at a stationary point. This reveals a very strong rigidity in the local structure of any such functional. As a corollary, this gives a new proof of the results of Okikiolu on local maxima and minima for the determinant of the conformal Laplacian, and we obtain results of the same type in general examples.

U2 - 10.1093/imrn/rnt122

DO - 10.1093/imrn/rnt122

M3 - Journal article

SN - 1073-7928

VL - 2014

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 22

ER -