Applications of the Ashtekar gravity to four dimensional hyperk\"ahler geometry and Yang-Mills Instantons

The Ashtekar-Mason-Newman equations are used to construct the hyperk\"ahler metrics on four dimensional manifolds. These equations are closely related to anti self-dual Yang-Mills equations of the infinite dimensional gauge Lie algebras of all volume preserving vector fields. Several examples of hyperk\"ahler metrics are presented through the reductions of anti self-dual connections. For any gauge group anti self-dual connections on hyperk\"ahler manifolds are constructed using the solutions of both Nahm and Laplace equations.