Some Fully Nonlinear Equations In Complex Geometry

Fully nonlinear elliptic equations play an important role in geometric analysis.We study Fu-Yau equations and deformed Hermitian-Yang-Mills equations.To study com-pactification of superstring,the Strominger system was introduced.This system can be viewed as a generalization of Calabi’s problem for non-Kahler complex manifolds.To solve it,Fu-Yau reduced this system to the Fu-Yau equation.On the other hand,it is an important problem to find canonical metrics in complex geometry.The solution of the Strominger system have some canonical properties.Motivated by Mirror Symme-try and Mathematical Physics,the deformed Hermitian-Yang-Mills equation has been studied extensively.By solving it,on the holomorphic line bundle,we can find a Hermi-tian metric such that the angle on its curvature is constant.We study a priori estimates,existence and uniqueness of these equations.We also prove some properties of met-rics on Calabi-Eckmann manifolds and the stability of harmonic maps from torus to Calabi-Eckmann manifolds.My work is as follows:1.We study the Fu-Yau equation in higher dimensions and prove the existence of solutions for Fu-Yau equations.The difficulty is that we need to establish a non-degeneracy estimate.As an application,we prove the existence of solutions for the modified Strominger system on Goldstein-Prokushki manifolds.2.We study the Fu-Yau equation and prove the existence of solutions on astheno-Kahler manifolds.We also prove the uniqueness of solutions for Fu-Yau equa-tions when the slope parameter a is negative.For α>0,we prove the uniqueness of solutions in a set which the norm of the solutions satisfy some conditions.3.We consider the deformed Hermitian-Yang-Mills equation on closed almost Her-mitian manifolds.In the case of hypercritical phase,we derive a priori estimates under the existence of an C-subsolution.We also proved the existence of solu-tions for the deformed Hermitian-Yang-Mills equation by assuming the condition of existence of a supersolution.As an application,we find a metric on complex line bundle such that the angle on the(1,1)part of its curvature is constant.4.We construct a family of Hermitian metrics on Calabi-Eckmann manifold S3 × S3 and study their adiabatic limits.We also give a family of Hermitian metrics on the Hopf surface S3 × S1,whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group.These metrics are locally conformally Kahler.Among the toric fibres of π:S2n-1 × S1→CP1,n of them are stable minimal surfaces and each of the n has a neighbourhood so that fibres therein are given by stable harmonic maps from n-torus and outside,far away from n tori,there are unstable harmonic ones that are also unstable minimal surfaces.