Fully non-linear elliptic equations on compact Hermitian manifolds
The author the author considers fully nonlinear elliptic equations on compact Hermitian manifolds \((M,\alpha)\). Let \(\chi\) be a fixed \((1,1)\)-form on \(M\). For a \(C^2\) function \(u:M\rightarrow M\) and \(g=\chi+i\partial\bar{{\partial}}u\) let \(A=\alpha^{i\bar{p}}g_{j\bar{p}}\) be the endomorphism of \(T^{(1,0}M\) which is a Hermitian endomorphism with respect to the metric \(\alpha\). The author considers the equation for u of the form \[F(A)=h,\tag{\(*\)}\] where \(h\) is a given function on \(M\) and \(F(A)=f(\lambda_1,\lambda_2,\dots,\lambda_n)\) is a smooth, symmetric function of the eigenvalues of \(A\). The author assumes that \(f\) is defined on an open, symmetric cone \(\Gamma\subset \mathbb R^n\) with vertex at the origin, containing the positive orthant \(\Gamma_n\). A function \(\underline{u}\) is called a \( C\)-subsolution of \((*)\) if for each \(x\in M\) the set \(\{\lambda'\in\Gamma:f(\lambda')=h(x) \text{ and } \lambda'-\lambda(B(x))\in \Gamma_n\}\) is bounded, where \(\lambda(B(x))\) denotes the \(n\)-tuple of eigenvalues of \(B(x), B=B^i_j=\alpha^{i\bar{p}}(\chi_{j\bar{p}}+\partial_j\partial_{\bar{p}})\). The main result is the following: Theorem. Suppose that \(u\) is a solution and \(\underline{u}\) is a \( C\)-subsolution of \((*)\). If we normalize \(u\) so that \(\sup_Mu=0\), then we have the estimate \(||u||_{C^{2,\alpha}}<C\), where \(C\) depends on the given data \(M,\alpha,\chi,h\) and the subsolution \(\underline{u}\). The author applies the result to the class of Hessian quotient equations on Kähler manifolds.
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