Volume growth of submanifolds and the Cheeger isoperimetric constant

Abstract

Abstract. We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above.

On the entropies of hypersurfaces with bounded mean curvature

“…This inequality combined with Lemma 5.2 gives the generalization to p-Laplacian of the Mc Kean inequality obtained in [33], and combined with Lemma 5.3 gives the inequality λ 0,p (M ) ≥ ((n − 1) k − a) p p p which generalizes the aformentioned results of [10] and [5] 4. Theorem A in [23] and the main Theorem in [22] can be immediately deduced from our results. Let us be more precise.…”

“…which generalizes the aformentioned results of [10] and [5] 4. Theorem A in [23] and the main Theorem in [22] can be immediately deduced from our results. Let us be more precise.…”

“…• By Lemma 5.1 and the upper bound in (23), if Ric M ≥ −(m − 1) k 2 , then we obtain the well known Mac Kean estimate:…”

“…For the second inequality, it suffices to observe that for any x ∈ M , R > 0 and any N satisfying the hypothesis, one has (23) follows from the definition of the entropies.…”

“…Let us be more precise. In [23], V. Gimeno and V. Palmer proved that if M is a minimal m-dimensional submanifold properly immersed in a simply connected manifold N , such that…”