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An a priori estimate for a fully nonlinear equation on four-manifolds. (English) Zbl 1067.58028
Let \((M,g)\) be a 4-dimensional Riemannian manifold without boundary and let \(W\), Ric, \(R\) and \(A\) denote, respectively, the Weyl curvature tensor, Ricci curvature, scalar curvature of \(g\) and \(A=A_g=\text{Ric}-\frac{1}{6}Rg\). For a metric \(g_0\), the corresponding quantities defined by \(g_0\) are denoted by attaching a sub- or superscript \(0\). For \(g=e^{2\omega}g_0\), \[ \sigma_k(A_g)=\sigma_k(A_0-2\nabla_0^2 \omega+2d\,\omega\otimes d\,\omega-| d\,\omega| ^2g_0), \] where \(\sigma_k\) denotes the \(k\)th elementary symmetric polynomial, applied to the eigenvalues of \(A_g\). The equation \[ \sigma_k(A_g)=f\tag{*} \] was introduced by J. A. Viaclovsky [Duke Math. J. 101, 283–316 (2000; Zbl 0990.53035)] and the present authors [Ann. Math. 155, 711–789 (2002)] prove that any metric \(g_0\) satisfying \[ \int\sigma_2(A_{g_0})d\text{vol}_{g_0}> 0,\quad Y(g_0) >0, \tag{**} \] where \(Y(g_0)\) is the Yamabe invariant of \(g_0\), is conformal to a metric \(g\) with \(\sigma_2(A_g)>0\). As the main existence result the authors prove that for any positive (smooth) function \(f\) there exist a solution \(g=e^{2\omega}g_0\) of (*) if \(g_0\) satisfies (**).

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Zbl 0990.53035
Full Text: DOI
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