Degeneration of Riemannian metrics under Ricci curvature bounds

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Laplacian cut-offs, porous and fast diffusion on manifolds and other applications (2016, methodology): “In the almost Euclidean case where α = 2, we actually use a construction inspired by [9] which relies on the fact that the Laplacian of the distance function satisfies the weak inequality ∆r ≤ Cr−1 globally on M , and allows to construct cut-offs which are 1 on the ball of radius R and vanish off in”
On the structure of almost einstein manifolds (2012, methodology): “Calculating Δ(χ|∇ log u|(2)) and applying the maximum principle, as done in [11], we obtain”
Space of Ricci Flows I (2009, background): “Then we return to the smooth manifold case, which is discussed clearly in [15].”
Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions (2013, methodology): “ƒ i /: Following notation from [10], we introduce the function LR.”
Asymptotically conical Calabi-Yau manifolds, III (2012, background): “Cheeger-Colding theory [16] implies that each blowdown sequence (M,λig), λi → 0, subconverges in the pointed Gromov-Hausdorff sense to the cone over some length space.”