On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

Four-dimensional closed manifolds admit a weak harmonic Weyl metric / G. Catino, P. Mastrolia, D. Monticelli, F. Punzo. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - (2022), pp. 2250047.1-2250047.34. [Epub ahead of print] [10.1142/S021919972250047X]

Four-dimensional closed manifolds admit a weak harmonic Weyl metric

P. Mastrolia
Secondo
;
2022

Abstract

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.
Canonical metrics; four manifolds; weak harmonic Weyl metrics; Einstein metrics;
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
ago-2022
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/940553
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