From the text: We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. All these equations are of the form \[ F(g^ {-1}W)=f(x,u)h(x,\nabla u), \] where \(g^ {-1}\) is the inverse of the metric tensor \(g\) on the manifold, \(W\) is a \((0,2)\) tensor given by \[ W=\nabla^ 2 u+a(x)du\otimes du+b(x)| \nabla u| ^ 2g + B(x), \] \(a\), \(b\), \(B\), \(f\), \(h\) are functions of the corresponding variables and \(F\) is a homogeneous symmetric function of degree one defined on an open convex cone \(\Gamma\) with \(\{\lambda\,\colon \,\lambda_ i>0,\,\forall i\}\subset\Gamma\subset\{\lambda\,\colon \, \sum_ i\lambda_ i>0\}\) normalized so that \(F(1,\ldots,1)=1\) and satisfying the following structure conditions: \(F\) is positive; \(F\) is concave; \(F\) is monotone.
The advantage of our method is that we derive Hessian estimates directly from \(C^0\) estimates. Also, the method is flexible and can be applied to a large class of equations.