The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere

Abstract

Abstract. We characterize the standard S 3 as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: 4π. As a corollary we answer in the affirmative an interesting special case of a conjecture of M. Min-Oo's on the scalar curvature rigidity of the upper hemisphere.

Metrics of positive scalar curvature and unbounded min-max widths

“…If g has positive Ricci curvature and scalar curvature R ≥ 6, the 4π upper bound on the supremum of the left-hand-side of the above expression was previously obtained by Eichmair [12]. We observed that the supremum of the isoperimetric profiles of the examples that we use also form an unbounded sequence.…”

“…Since the proof of the positive mass conjecture in general relativity by Schoen and Yau [23], and Witten [26], the rigidity phenomena involving the scalar curvature has been fascinating the geometers. These results play an important role in modern differential geometry and there is a vast literature about it, see ( [1,3,4,5,6,7,8,12,15,16,17,18,19,22]). Many of these works concern rigidity phenomena involving the scalar curvature and the area of minimal surfaces of some kind in three-manifolds.…”

Rigidity of min-max minimal spheres in three-manifolds

“…It is well-known that lower bounds on the scalar curvature of M give some information on the space of minimal surfaces. Several rigidity theorems have been obtained assuming the existence of an area-minimizing surface of some kind ( [3], [4], [7], [13], [25], [34]), but no known result asserts rigidity under the presence of a minimal surface produced by min-max methods. In this paper we prove theorems in that direction.…”

Rigidité conforme des hémisphères $${{\rm S}^4_+}$$ et $${{\rm S}^6_+}$$

“…Cependant, dans un travail récent [1], Brendle, Marques et Neves construisent, pour n ≥ 3, des métriques sur l'hémisphère à courbure scalaire strictement supérieure à n(n − 1) satisfaisant les conditions (2) et (3). De tels exemples mettent donc en défaut la validité de l'énoncé de Min-Oo.…”

“…Dans [10], Hang et Wang montrent qu'en remplaçant l'hypothèse (1) sur la courbure scalaire par son analogue sur la courbure de Ricci, la conjecture est vérifiée. Le cas de la dimension 3 est traité dans [3] avec des conditions supplémentaires sur le profil isopérimétrique de ∂ M. Dans un autre travail [9], Hang et Wang s'intéressent aux déformations conformes des métriques sur l'hémisphère standard et ils prouvent en particulier la conjecture pour ce type de transformations. Plus précisément, ils montrent: Théoréme 1 ([9]) Soit g = e 2u g st une métrique dans la classe conforme de la métrique ronde g st à courbure scalaire R g ≥ n(n − 1).…”