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Mean curvature in the light of scalar curvature. (Courbure moyenne à la lumière de la courbure scalaire.) (English. French summary) Zbl 1467.53041

A Riemannian manifold \(X\), with boundary \(\partial X=Y\) in a larger Riemannian manifold, is called mean convex if the boundary \(Y\) has positive mean curvature. Scalar curvature can be related to mean curvature, for example let \(X\) be a domain in \({\mathbb R}^n\), it is known that the natural \(C^0\)-Riemannian metric on the double \(X\times _Y X\) with \(Y=\partial X\) of positive mean curvature can be approximated by smooth metrics with positive scalar curvature. The main purpose of this paper is to search for constraints on global geometric invariants of \(X\) such that \(\operatorname{Scal}(X)>0\) that generalise theorems on smooth mean convex domains in the Euclidean spaces. Moreover the author finds that techniques used for the study of manifolds with scalar curvature greater or equal than a positive constant give results for hypersurfaces in \({\mathbb R}^n\) with mean curvature greater or equal than a positive constant. In this framework several conjectures are formulated and some theorems are proved.

MSC:

53C20 Global Riemannian geometry, including pinching
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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