WIDTH, RICCI CURVATURE, AND MINIMAL HYPERSURFACES PARKER GLYNN-ADEY AND YEVGENY LIOKUMOVICH

Abstract. Let (M, g0) be a closed Riemannian manifold of dimension n, for 3 ≤ n ≤ 7, and non-negative Ricci curvature. Let g = φ2g0 be a metric in the con-formal class of g0. We show that there exists a smooth closed embedded minimal hypersurface in (M, g) of volume bounded by CV n−1 n, where V is the total volume of (M, g) and C is a constant that depends only on n. When Ric(M, g0) ≥ −(n−1) we obtain a similar bound with constant C depending only on n and the volume of (M, g0). Our second result concerns manifolds (M, g) of positive Ricci curvature. We obtain an effective version of a theorem of F. Coda Marques and A. Neves on the existence of infinitely many minimal hypersurfaces on (M, g). We show that for any such manifold there exists k minimal hypersurfaces of volume at most CnV sysn−1(M)) − 1n−1 k 1n−1, where V denotes the volume of (M, g0) and sysn−1(M) is the smallest volume of a non-trivial minimal hypersurface. 1