Conformal energy in four dimension
In [Invent. Math. 69, 269-291 (1982; Zbl 0503.53042)], \textit{P. Li} and \textit{S.-T. Yau} introduced the notion of conformal volume for compact Riemannian manifolds which admit conformal immersions into spheres. This conformal volume depends only on the conformal class of the metric. In the same spirit, the authors of this article restrict to four-dimensional compact Riemannian spaces and define their conformal energy using the Paneitz operator. This is a fourth order elliptic operator involving the Laplacian and the curvature of the manifold. Again, this notion only depends on the conformal structure. Two lower bounds for the conformal energy are derived: one in terms of the first non-zero eigenvalue of the Paneitz operator and another in terms of the conformal volume. Calculating the conformal energy of a given compact four-manifold is not an easy task. The authors succeed in doing this for \(S^2\times S^2\). Finally, they derive an obstruction to the existence of minimal immersions of compact four-manifolds into spheres.
scientific article; zbMATH DE number 1889184