The modified Yamabe problem and geometry of modified scalar curvatures
In this paper, \((M,g)\) is a compact oriented \(n\)-manifold, \(n\geq 3\), equipped with a smooth metric \(g\) with scalar curvature \(s_g\) and Weyl conformal curvature \(W_g\). The modified Yamabe problem is a natural extension of the classical Yamabe problem proposed by \textit{M. Gursky} in his paper [Math. Ann. 318, No. 3, 417--431 (2000; Zbl 1034.53032)]. It is concerned with a modified scalar curvature which satisfies a conformal covariance condition. The problem is whether there exists a smooth metric \(\tilde g\) conformal to \(g\) such that this modified scalar curvature is constant. The modification adopted in this paper is a combination of the scalar curvature and the Weyl conformal curvature \(W_g\), depending on a certain real parameter \(t\), with \(t=0\) corresponding to the original Yamabe problem. The author proves an existence theorem analogous to a theorem of Trudinger and of Aubin [see \textit{N. Trudinger}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22, 265--274 (1968; Zbl 0159.23801) and \textit{T. Aubin}, J. Math. Pur. Appl., IX. Sér. 55, 269--296 (1976; Zbl 0336.53033)]. The author then obtains new results in dimension four by using modified curvature of a certain type and interpreting it geometrically. Thus, he obtains an extension of the four-dimensional sphere theorem of \textit{M. Micallef} and \textit{J. Moore} [Ann. Math. (2) 127, No. 1, 199--227 (1988; Zbl 0661.53027)]. Another modified scalar curvature is used in solving a problem concerning self-dual Einstein 4-manifolds, asking whether a compact negative Ricci, Einstein self-dual 4-manifold is either real or complex hyperbolic. The author also obtains results towards the Goldberg conjecture, which states that if a compact, almost Kähler \(2n\)-manifold is Einstein, then it must be Kähler. The result obtained, using a modified scalar curvature, is again under negative Ricci condition.
scientific article; zbMATH DE number 2183023
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