Lipschitz convergence of Riemannian manifolds

Compact 8-manifolds with holonomy Spin(7)

“…From (6) we deduce that if dΩ 10 t −1/5 , and φ 2 t 4 , then η is small in C 0 . This estimate on η C 0 is the important ingredient needed to make the proof work.…”

“…Therefore, we are able to prove the following result. Suppose that φ is a 4-form with dΩ + dφ = 0, that t > 0, that δ(g) ≥ t and R(g) C 0 ≤ t −2 , and that dΩ 10 t −1/5 and φ 2 t 4 . Then there exists a Spin(7)-structureΩ near to Ω with dΩ = 0.…”

“…Substituting r = Lt into (59), raising the whole inequality to the power 1/10 and manipulating shows that ∇χ 10 Finally, taking the supremum of (55) over all m ∈ M and substituting r = Lt we find that…”

Pinching, Pontrjagin classes, and negatively curved vector bundles

“…Note that if X k is a complete Riemannian manifold for all k , then the space X is necessarily a C ∞ -manifold with a complete C 1,α -Riemannian metric [GW88]. If each X k is a Hadamard manifold, then for any x k ∈ X k , the sequence (X k , x k ) is precompact in the pointed Lipschitz topology because the injectivity radius of X k at x k is uniformly bounded away from zero [Fuk86,p.132].…”

A Convergence Theorem for Riemannian Submanifolds

“…We refer to, for example, [Ch,NI,N2,GI,Ps,GW,K] for details. M. Anderson has extended this theorem to a larger class of Riemannian manifolds (see [AM]).…”