On a class of fully nonlinear elliptic equations on Hermitian manifolds
On a compact \(n\)-dimensional Hermitian manifold \((M, \omega )\) with smooth boundary, consider a smooth real \((1,1)\)-form \(\chi \) and the equation for the unknown \(u\) (such that \(\chi +dd^c u >0\)) \[ (\chi +dd^c u)^n =f (\chi +dd^c u)^{n-k} \wedge \omega ^k, \] for \(1\leq k\leq n\). For \(k=1\), if \(M\) is a compact Kähler manifold, \(\chi\) is Kähler and \(f\) constant, the equation is satisfied by a stationary point of the J-flow first studied in [\textit{S. K. Donaldson}, Asian J. Math. 3, No. 1, 1--15 (1999; Zbl 0999.53053)] and [\textit{X. Chen}, Int. Math. Res. Not. 2000, No. 12, 607--623 (2000; Zbl 0980.58007)]. For general \(k\), the equation was introduced in [\textit{H. Fang} et al., J. Reine Angew. Math. 653, 189--220 (2011; Zbl 1222.53070)]. The main result of the present paper says that, for smooth \(f\) and smooth boundary condition \(\phi\), if there exists a \(C^2\) (subsolution) \(v\) such that \[ (\chi +dd^c v)^n \geq f (\chi +dd^c v)^{n-k} \wedge \omega ^k \] and \(v=\phi\) on \(\partial M\), then a solution exists as well. The main ingredient of the proof are \(C^2\) estimates generalizing those from the last cited paper to the Hermitian setting.
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