Metrics of positive scalar curvature and unbounded min-max widths

Abstract

In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to 6 and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar curvature metrics due to Gromov and Lawson. We develop analogies between the area of boundaries of special open subsets in our three-manifolds and 2-colorings of associated full binary trees. Then, via combinatorial arguments and using the relative isoperimetric inequality, we argue that the widths converge to infinity.

Min-max minimal surfaces, horizons and electrostatic systems

“…From the physical point of view we may think of this condition as an obstruction to the existence of black holes. Moreover, as shown in [65], the bound (74) on the width is no longer true if one only assume R g ≥ 2Λ, but allow the presence of stable minimal spheres.…”

Waist inequality for 3-manifolds with positive scalar curvature

“…Note that it is not possible to obtain a version of Theorem 1.1 with area or diameter of the connected component of f −1 (t) replaced by area or diameter of the whole fiber f −1 (t). Indeed, one can construct manifolds that are hard to cut by considering Gromov-Lawson connect sums of 3-spheres corresponding to large trees or expander graphs (see [Mon15], [PS19]).…”

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

“…Related results, and structure of the paper. In the aforementioned work [24], Marques and Neves proved sharp estimates for the width of three-spheres with positive Ricci curvature in terms of a lower bound for the scalar curvature (the hypothesis on the Ricci curvature cannot be weakened to a hypothesis on the scalar curvature, see [33]). Their proof involves the analysis of how does the Simon-Smith width evolve under the Ricci flow.…”