In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map φ:M→N. The resulting system, denoted by (φ-CPE), is natural from the variational viewpoint and describes stationary points for the integrated φ-scalar curvature functional restricted to metrics with unit volume and constant φ-scalar curvature. We prove both a rigidity statement for solutions to (φ-CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (φ-CPE) with φ a harmonic map.

Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi$$-CPE Metric in Its Conformal Class / G. Colombo, L. Mari, M. Rigoli. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 32:11(2022 Nov), pp. 267.1-267.32. [10.1007/s12220-022-01000-3]

### Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi$$-CPE Metric in Its Conformal Class

#### Abstract

In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map φ:M→N. The resulting system, denoted by (φ-CPE), is natural from the variational viewpoint and describes stationary points for the integrated φ-scalar curvature functional restricted to metrics with unit volume and constant φ-scalar curvature. We prove both a rigidity statement for solutions to (φ-CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (φ-CPE) with φ a harmonic map.
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CPE metric; Besse conjecture; Vacuum static space; Scalar field; Harmonic map; Wave map
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/936254
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