On $3$-manifolds with small mass and $L^2$-curvature

Abstract

One of S.T. Yau's problems asks the following: given a $3$-dimensional asymptotically flat manifold $M$ with non-negative scalar curvature and $L^2$-norm of the curvature tensor at most $1$, if the mass of $M$ is small, is there a bilipschitz diffeomorphism from $M$ to the flat Euclidean space ${\mathbb{R}}^3$? We provide a strong positive answer to this problem by using our previous work \cite{DS25}.

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