Rigidity of min-max minimal spheres in three-manifolds

Abstract

Abstract. In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than 4π and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

Applications of Almgren-Pitts Min-max theory

“…The proof can be found in [20,Section 5] or [30,Section 5]. In Lemma 5.3 of [30] we show that the Morse function f can be chosen so that, for every k ∈ N, each level set f −1 (t) intersects c(k) in at most one point and no critical point of f belongs to c(k).…”

“…In Lemma 5.3 of [30] we show that the Morse function f can be chosen so that, for every k ∈ N, each level set f −1 (t) intersects c(k) in at most one point and no critical point of f belongs to c(k). Thus, we can find ε sufficiently small so that for all t ∈ R and…”

“…We now describe the idea to exclude the case L = ω p (M ) and obtain a contradiction. The argument in [30,Section 6] requires the use of the discrete theory of Almgren-Pitts and the reader can see there the details.…”

Width, Ricci curvature, and minimal hypersurfaces

“…Other important results are contained in a paper of Marques and Neves [25] where, among other things, they prove a sharp upper bound on W (M ), when M is a Riemannian 3-sphere with Ricci > 0 and scalar curvature R ≥ 6.…”

Min-max theory, Willmore conjecture and the energy of links

“…For instance, it is expected that if we replace the one-dimensional sweepouts by k-parameter families of surfaces then the Morse index of the minmax minimal surface should be bounded above by k. (See [18] and [32] for the particular case of positive Ricci curvature and k = 1.) Although the Almgren-Pitts theory successfully applies to any number of parameters, the general bound on the index is not known.…”