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We prove that generalised Monge-Ampere equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampere equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation, and prove a conjecture of Szekelyhidi in the projective case on the solvability of inverse Hessian equations. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.
We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case,
In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curv
In this paper we consider the Monge-Amp`{e}re type equations on compact almost Hermitian manifolds. We derive a priori estimates under the existence of an admissible $mathcal{C}$-subsolution. Finally, we also obtain an existence theorem if there exists an admissible supersolution.
A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercom
We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equati