On A Class Of Fully Nonlinear Equations Over Almost Hermitian Manifolds And HKT Manifolds

In the current thesis we are mainly focus on the study of a class of fully non-linear partial differential equations on manifolds.All such equations including Monge-Ampere type equations,special Lagrangian equations,Hessian equations etc,often have a closely connection with mathematics and physics.The present thesis consists of five chapters.In chapter 1,we are mainly recall some new developements of fully nonlinear partial differential equations in real mani-folds,almost complex manifolds and hypercomplex manifolds.In chapter 2,we study the Monge-Ampere type equations on almost Hermitian manifolds.Under the exis-tence of C-subsolutions,we obtain the second order estimates.And then by virtue of C2,?-estimates of Tosatti-Wang-Weinkove-Yang[82]and the Schauder theory for ellip-tic equations,we can also get the higher order estimates.In addition,under the existence of supersolutions,the existence result can be also concluded by using the continuity method.In chapter 3,we study the parabolic Monge-Ampere type equations.Using the estimates of elliptic equations in chapter 2,we give a new of the existence result of the Monge-Ampere type equations in Chapter 2.In chapter 3,based on the hypercrit-ical phase case which was studied by us[53],we also consider the special Lagrangian equation with supercritical phase,and we obtain the gradient estimates by using maxi-mum principle.In chapter 5,we study the fully nonlinear partial differential equations on the HKT manifolds which admits a locally flat hyperKahler metric.Under the exis-tence of C-subsolutions,we obtain the second order estimates.We further develop the Evans-Krylov theory in this setting,along with Schauder theory,these yield the higher order estimates.As an application,we also study the existence results for some special equations.