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On static manifolds and related critical spaces with zero radial Weyl curvature

Emanuel Mendonça Viana

Instituto Federal de Educação, Ciência e Tecnologia do Ceará - IFCE, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak025746ecd810ccd9566a0e1d4a082706').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy025746ecd810ccd9566a0e1d4a082706 = '&#101;m&#97;n&#117;&#101;l.m&#101;nd&#111;nc&#97;' + '&#64;'; addy025746ecd810ccd9566a0e1d4a082706 = addy025746ecd810ccd9566a0e1d4a082706 + '&#105;fc&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'br'; var addy_text025746ecd810ccd9566a0e1d4a082706 = '&#101;m&#97;n&#117;&#101;l.m&#101;nd&#111;nc&#97;' + '&#64;' + '&#105;fc&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'br';document.getElementById('cloak025746ecd810ccd9566a0e1d4a082706').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy025746ecd810ccd9566a0e1d4a082706 + '\'>'+addy_text025746ecd810ccd9566a0e1d4a082706+'<\/a>';

The aim of this paper is to study compact Riemannian manifolds $(M,\,g)$ that admit a non-constant solution to the system of equations $$-\Delta f\, g+Hess f-fRic=\mu Ric+\lambda g,$$ where $Ric$ is the Ricci tensor of $g$ whereas $\mu$ and $\lambda$ are two real parameters. More precisely, under the assumption that $(M,\,g)$ has zero radial Weyl curvature, this means that the interior product of $\nabla f$ with the Weyl tensor $W$ is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.

The article can be found at https://doi.org/10.1007/s00605-019-01365-8

Joint work with Abdênago Alves de Barros (Universidade Federal do Ceará - UFC, Departamento de Matemática), Halyson Irene Baltazar (Universidade Federal do Piauí - UFPI, Departamento de Matemática) and Rondinelle Marcolino Batista (Universidade Federal do Piauí - UFPI, Departamento de Matemática).