Critical metrics of the total scalar curvature functional on 4‐manifolds. Issue 16 (24th June 2015)

• Record Type: Journal Article
• Authors: Barros, A.; Leandro, B.; Ribeiro, E.
• Is Part Of: Mathematische Nachrichten. Volume 288:Issue 16(2015)
• Page Start: 1814
• Page End: 1821
• Publication Date: 2015-06-24
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201400390
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980's that every CPE metric must be Einstein. We prove that a 4‐dimensional CPE metric with harmonic tensor <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkz9r456d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201400390:mana201400390-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>W</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math></alternatives></inline-formula> must be isometric to a round sphere <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkz9r457x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201400390:mana201400390-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="double-struck">S</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></p> </abstract>