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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 (**).

##### MSC:
 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
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##### References:
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