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Global models of Riemannian metrics.

Juan Fontanillas, Fernando Varela (1987)

Revista Matemática Iberoamericana

In this paper we give certain Riemannian metrics on the manifolds Sn-1 x S1 and Sn (n ≥ 2), which have the property to determine these manifolds, up to diffeomorphisms.The global expressions used for Riemannian metrics are based on the global expression for exterior forms studied in [4]. In [3] one finds certain metrics using global expressions that differ from the type we propose.To some extent, Theorem 3 is a generalization for metrics in an arbitrary dimension, of a theorem proved in [2] for...

Global pinching theorems for minimal submanifolds in spheres

Kairen Cai (2003)

Colloquium Mathematicae

Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere S n + p ( 1 ) . By using the Sobolev inequalities of P. Li to get L p estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and | | σ | | p the mean curvature and the L p norm of the square length of the second fundamental form of M. We show that there is a constant C such that if | | σ | | n / 2 < C , then M is a minimal submanifold in the sphere S n + p - 1 ( 1 + H ² ) with sectional curvature...

Growth of a primitive of a differential form

Jean-Claude Sikorav (2001)

Bulletin de la Société Mathématique de France

For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function f , by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if M has bounded geometry. For a volume form, it suffices to have the inequality ( | Ω | ≤ ∫ ∂ Ω f d σ for every compact domain Ω ⊂ M ). This implies in particular the “well-known” result that if M is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then the volume...

Growth of weighted volume and some applications

Mirjana Milijević, Luis P. Yapu (2020)

Archivum Mathematicum

We define cut-off functions in order to allow higher growth for Dirichlet energy. Our results are generalizations of the classical Cheng-Yau’s growth conditions of parabolicity. Finally we give some applications to the function theory of Kähler and quaternionic-Kähler manifolds.

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