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

G. Colombo;M. Rigoli
2022

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.
CPE metric; Besse conjecture; Vacuum static space; Scalar field; Harmonic map; Wave map
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
Article (author)
File in questo prodotto:
File Dimensione Formato  
CPE_Besse_revised_12_7.pdf

accesso aperto

Tipologia: Pre-print (manoscritto inviato all'editore)
Dimensione 818.52 kB
Formato Adobe PDF
818.52 kB Adobe PDF Visualizza/Apri
s12220-022-01000-3.pdf

accesso riservato

Tipologia: Publisher's version/PDF
Dimensione 468.8 kB
Formato Adobe PDF
468.8 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/936254
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact