Weighted Sobolev Inequalities and Ricci Flat Manifolds

Improved Hardy and Rellich inequalities on Riemannian manifolds

“…In [14], Grillo obtained Hardy, Rellich and Sobolev inequalities in the context of homogeneous spaces. Similarly, in [19], [20], [9], geometric properties such as volume growth and isoperimetric profile are related to the existence of Hardy type inequalities. A detailed treatment of improved Hardy inequalities with best constants, involving various kinds of distance functions in the Euclidean space R n , can be found in [4].…”

Rigidity for Multi-Taub-NUT metrics

“…In view of [Mi1], under the other assumptions, the cubic curvature decay is indeed automatic as soon as the curvature decays faster than quadratically. And it implies the covariant derivatives obey ∇ i Rm = O(ρ −3−i ) [Mi2].…”

Weighted Sobolev inequalities under lower Ricci curvature bounds

“…This quickly yields a global Sobolev inequality under lower Ricci bounds (Corollary 2.8), which implies Gallot's inequality [6] on compact manifolds. In Section 2.3, we then combine Corollary 2.6 with patching methods from Grigor'yan and Saloff-Coste [7] and Minerbe [16], and with the Cheeger-Colding segment inequality [4], to prove Theorem 1.2. …”

“…The novelty in our approach here is that we revisit foundational work on isoperimetry due to Gromov [8] to first prove a sharp Sobolev inequality with Dirichlet boundary conditions on annuli (Corollary 2.6(ii)). We can then apply a patching scheme based on [7] and [16]. Recent, independent work due to van Coevering [21] contains a proof of (1.3) via scaling and patching.…”

“…[19] obtain an estimate such as (1.1) in a rather more flexible setting, but with weaker exponents that may depend on lower bounds for |B(x, 1)|, x → ∞, and (1.2) overlaps with results of Minerbe [16] for Ric ≥ 0, one of whose methods, originating from [7], we borrow. The novelty in our approach here is that we revisit foundational work on isoperimetry due to Gromov [8] to first prove a sharp Sobolev inequality with Dirichlet boundary conditions on annuli (Corollary 2.6(ii)).…”