Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4‐manifolds

Abstract

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds.In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded. K E Y W O R D S 4-manifold, biorthogonal curvature, minimal volume, sphere theorem M S C ( 2 0 1 0 ) Primary: 53C21, 53C23; Secondary: 53C25This concept plays an important role in geometric topology. It is closely related with other important invariants as, for instance, minimal entropy h(M) and simplicial volume ‖ ‖. Paternain and Petean [37] proved that the minimal volume, on a compact manifold , satisfies the following chain of inequalitieswhere ( ) is a positive constant; for more details, we refer the reader to [23,24] and [37]. It should be emphasized that some authors have studied other minimal volume invariants in a similar context (cf. [30,45] and [46]). Among them, let us highlight the following ones: Gromov minimal volume, which is defined by

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

“…Recently, Ribeiro and Costa [10] used the notion of biorthogonal (sectional) curvature to obtain some interesting sphere theorems. In particular, they were able to improve the pinching constants obtained in some previous works.…”

“…We proceed to state our results. Indeed, motivated by the ideas developed by Ribeiro and Costa [9,10], Gu and Xu [13] as well as Xu and Zhao [25], we shall investigate sphere theorems for submanifolds under certain conditions involving the biorthogonal (sectional) curvature. After these preliminary remarks, we may announce our first result as follows.…”