Width, Ricci curvature, and minimal hypersurfaces

Abstract

Abstract. Let (M, g0) be a closed Riemannian manifold of dimension n, for 3 ≤ n ≤ 7, and non-negative Ricci curvature. Let g = φ 2 g0 be a metric in the conformal class of g0. We show that there exists a smooth closed embedded minimal hypersurface in (M, g) of volume bounded by C(n)V n−1 n , where V is the total volume of (M, g). 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 and dimension at most seven. We obtain an effective version of a theorem of F. C. 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 sys n−1 (M ), where V denotes the volume of (M, g0) and sys n−1 (M ) is the smallest volume of a non-trivial minimal hypersurface.

Weyl law for the volume spectrum

“…We consider a compact Riemannian (n + 1)-manifold (M n+1 , g) isometrically embedded in R Q with smooth boundary ∂M and satisfying (4). Recall the definition ofω k p (M ) in (7), that C denotes the unit cube in R n+1 , and that, for every integer 0 ≤ k < n + 1, we set a(n, k) = lim p→∞ω k p (C).…”

Families of short cycles on Riemannian surfaces

“…all expressions of the form k i=1 a i t i , where a i ∈ Z 2 and t i ∈ S 1 . For any 1-sweepout z t the family of cycles { k i=1 a i z t i } k i=1 a i t i ∈T P p (S 1 ) is a k-sweepout of M (see [14], [7]). We estimate the mass of each cycle l(…”

“…More recently, in [23] Marques and Neves used k-sweepouts to prove existence of infinitely many minimal hypersurfaces in manifolds of positive Ricci curvature. In [7] Glynn-Adey and the author obtained upper bounds for volumes of these hypersurfaces.…”

“…This corresponds to the case k = 1 of Theorem 1.1. Different proofs of their result, improving the value of an upper bound for the constant C, were given in [21], [7]. The proof of Balacheff and Sabourau relies on the estimate of Li and Yau [20] for the first eigenvalue of the Laplacian.…”

On the min-max width of unit volume three-spheres

“…W (S 3 , g) ≤ Cvol(S 3 , g) 2 3 . More recently, Glynn-Adey and Liokumovich [17] proved that, for every conformal class of Riemannian metrics on S 3 (actually, on any closed manifold), there exists a constant that bounds from above the Almgren-Pitts width of every unit volume metric within this class. The precise statement of their result is actually rather explicit about the geometric dependence of the constant they obtain, see Theorems 5.1 therein; in particular, their estimate implies that all unit volume positive Ricci curvature metrics have their Almgren-Pitts widths uniformly bounded from above as well.…”