Hermitian Yang–Mills equations and their generalizations

Hermitian Yang–Mills equation is a nonlinear equation on a Hermitian metric defined on a holomorphic vector bundle over a compact Kähler manifold. It can be also considered as an equation on the unitary connection associated with this Hermitian metric. If the dimension of the base manifold is equal to 1 then solutions of Hermitian Yang–Mills equation are given by flat connections. If this dimension is equal to 2 then such solutions are given by anti-selfdual connections called otherwise the instantons. So Hermitian Yang–Mills equations may be considered as a multi-dimensional generalization of the duality equations.
The main result of the first part of the talk, related to Hermitian Yang–Mills equations, is the Donaldson theorem on the existence and uniqueness of solutions of the boundary value Dirichlet problem for the Hermitian Yang–Mills equation on a compact Kähler manifold with boundary.
The second part is devoted to the deformed Hermitian Yang–Mills equation. This generalization of Hermitian Yang–Mills equation arose in the papers by Yau with coauthors. The deformed Hermitian Yang–Mills equation reduces to the Hermitian Yang–Mills equation in the large volume limit. The existence of solutions of the deformed Hermitian Yang–Mills equation under additional conditions of the positive curvature type is proved using the heat flow. This flow exists for all times and in the large volume limit converges to a solution of the deformed Hermitian Yang–Mills equation.