We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of S 1 inside S m , m ≥ 5, that are conformal to the round (incomplete) metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S m \ S 1 ∼ = S m−2 × H 2 .

Homoclinic classes of generic C 1 -diffeomorphisms are maximal transitive sets and pairwise disjoint. We here present a model explaining how two different homoclinic classes may intersect, failing to be disjoint. For that we construct a one-parameter family of diffeomorphisms (g s ) s∈[−1,1] with hyperbolic points P and Q having nontrivial homoclinic classes, such that, for s > 0, the classes of P and Q are disjoint, for s < 0, they are equal, and, for s = 0, their intersection is a saddle-node.

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In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.