No abstract

We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension n ≥ 3. The proof is based on the technique of Cheeger-Tian for Ricci-flat metrics. We also apply this method to obtain a singularity removal theorem for (extended) obstruction-flat metrics with isolated C 0 -orbifold singular points.

For a submanifold Σ ⊂ R N Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. More generally, on any metric measure space we are able to approximate a 1-parameter family of coarse Ricci functions that include the coarse Bakry-Emery Ricci curvature.

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the threeball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS], we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary.Let Ric(g s ) denote the Ricci curvature tensor of the metric g s . Then, as in [MNS, Section 6.5] and by Lemma 3.1, we haveMoreover, again by Lemma 3.1, we have ∂ s | s=0 Ric(g s ) = Hess g f + ∆ g f g > 0 on (−ε, 0] × D.So for s 1 sufficiently small, g s will have positive Ricci curvature on (−ε, 0]×D for all s ∈ [0, s 1 ].4 Since given any real number A, there exists a = a(A) > 0 such that for x ∈ (0, a], the function p(x) = exp −x −2 satisfies p ′′ (x) − Ap ′ (x) > 0. David J. Wraith, On the moduli space of positive Ricci curvature metrics on homotopy spheres, Geom. Topol. 15

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula