Gromov–hausdorff Convergence and Volumes of Manifolds

Let n ≥ 2, M and Mk (k = 1, 2, . . . ) be compact Riemannian nmanifolds, possibly with boundaries, and let {Mk} converge to M with respect to the Gromov–Hausdorff distance. We prove that Vol(M) ≤ lim infk→∞ Vol(Mk) provided that one of the following holds: (1) Mk are homotopy equivalent to M , and M admits either a nonzero-degree map onto the torus T n or an odd-degree map onto RP; (2) n = 2, and the Euler characteristics of Mk are uniformly bounded. For n ≥ 3 we give examples of convergence in which M and Mk are diffeomorphic to S and Vol(Mk) → 0.