Eigenvalue estimates for the -Laplace operator on manifolds

On the entropies of hypersurfaces with bounded mean curvature

“…The proof in [30] is done in the Euclidean case, but it can be easily adapted to a manifold. 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.…”

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

“…Geometric estimates on fundamental tones and the bottom of the spectrum of the Laplacian on Riemannian manifolds have been obtained in various contexts (see, e.g., [6,7,8,13,22] for p = 2 and φ constant, [16,23] for p ∈ (1, ∞) and φ constant) and for other results [1,2,17,18,21,25,33] and [3,14,19,22,24,32,34]. In this paper we present generalisations of some of these results to the setting of smooth metric measure spaces.…”

The $$p$$ p -hyperbolicity of infinity volume ends and applications

“…Namely, V (r) ≤ C exp(ar), where a = 4m in Theorem 5.1 (see [13]) and a = 2(2m + 1) in Theorem 5.2 (see [8]). Therefore, we get To proof the equality in the space form case we use Theorem 1.1 of [15] applied to the gradient of distance function. Precisely Here X(Ω) denotes the set of all smooth vector fields, X, on Ω with sup norm X ∞ = sup Ω X < ∞ (where X = g(X, X) 1/2 ) and inf Ω divX > 0.…”