EINSTEIN-METRICS WITH CUSPS AND SEIBERG-WITTEN EQUATIONS

If (M-4, g(0)) is a finite volume quotient of the complex hyperbolic s pace, we prove that any Einstein, complete, bounded curvature metric g on M, such that the diameter of horocycles goes to zero and the mean curvature of the horocycles is bounded from below by a positive consta nt, is equal (up to a diffeomorphism) to the standard complex hyperbol ic metric g(0). This generalizes a theorem of LeBrun in the compact ca se. To prove the theorem, we produce a solution of the Seiberg-Witten equations on the noncompact manifold (M,g); in fact, M can be compacti fied as an orbifold (M) over bar, and we get the wanted solution as a limit of solutions for a sequence of metrics on (M) over bar, which ap proximate g.