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In this paper, we discuss the Yang-Mills functional and a certain family ofits critical points on quantum Heisenberg manifolds using noncommutativegeometrical methods developed by A. Connes and M. Rieffel. In our main result,we construct a certain family of connections on a projective module over aquantum Heisenberg manifold that give rise to critical points of the Yang-Millsfunctional. Moreover, we show that this set of solutions can be described as aset of solutions to Laplace's equation on quantum Heisenberg manifolds.

The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds