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Local estimates for a class of fully nonlinear equations arising from conformal geometry. (English) Zbl 1042.53021

The \(\sigma _k\)-scalar curvature (\(k\geq 1\)) of a Riemannian manifold \((M, g)\) is defined as the \(k\)-th elementary symmetric function of the (eigenvalues of) the tensor field \(A_g = g^{-1}\cdot S_g\) defined locally by \((A_g)^i_j = g^{ik}(S_g)_{kj}\), where \(S_g = \frac{1}{n-2}\cdot (Ric_g - \frac{1}{2(n-1)}R_g\cdot g)\) (\(Ric_g\) and \(R_g\) being the Ricci tensor and scalar curvature of \(g\), \(n = \dim M\geq 3\)) is the Schouten tensor of \(g\). Deformations of \(\sigma _k\) in the conformal class of \(g\) yield the PDE (*) \(\sigma _k^{1/k}(\nabla ^2 u + du\otimes du - \frac{1}{2}\cdot \nabla u^2\cdot g + S_g) = \exp (-2u)\). The authors study the equation (**) arising from (*) by replacing its right hand side by \(f\cdot \exp (-2u)\), \(f\) being a positive function. The main result provides an estimate of \(\nabla u^2 + \nabla ^2 u\) for a solution \(u\) of (**) which gives a metric with \(\sigma _j (x) > 0\) for every \(j\leq k\) and \(x\in M\).

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
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