Bubble tree convergence for conformal metrics with ∫M|R|n2dVg bounds. (September 2020) - Xu, Ke
Abstract: Given a sequence of conformal metrics g k = u k 4 n − 2 g 0 on a smooth compact boundaryless Riemannian manifold ( M n, g 0 ) . Assume the volume of g k and L n 2 norm of scalar curvatures both are bounded. We prove that, after passing to a subsequence, u k weakly converges to the bubble tree limit ( u, u 1, 1, …, u i, α, …, u l, α l ), 1 ≤ i ≤ l < ∞, 1 ≤ α ≤ α i < ∞ in W 2, p, for some p < n 2, where u ∈ W 2, p ( M, g 0 ) and u i, α ∈ W 2, p ( R n, g R n ) . Moreover, after passing to a subsequence, the sequence of metric spaces ( M, d k ) defined by g k converges to a connected metric space ( M ∞, d ∞ ) in the Gromov-Hausdorff topology sense and lim k → ∞ Vol ( M, g k ) = H n ( M ∞, d ∞ ), where H n is the n dimensional Hausdorff measure defined by d ∞ .