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Four Lectures on Scalar Curvature
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Four Lectures on Scalar Curvature
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We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures
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Cited by 8 Pith papers
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A Positive Mass Theorem for Continuous Metrics
Proves that the harmonic mass of a continuous asymptotically flat metric on R^3 is non-negative, with equality only when the metric is flat.
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Scalar curvature under weak limits of manifolds
Scalar curvature lower bounds are preserved under weak limits of smooth closed 3-manifolds through μ-bubble comparisons when volumes and Lipschitz constants converge appropriately.
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Quantification of $C^0$ Convergence in Dimension Three
In three dimensions, infimum scalar curvature of a perturbed metric is at most the reference value at the origin plus C times the C0 norm to the power 1/2, for sufficiently small perturbations.
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Gromov's Euclidean Endpoint $C^0$ Rigidity for the Positive Mass Theorem
Proves that a smooth complete metric g on R^3 with Scal(g) >= 0 and |g - g_Euc| = o(r^{-1}) as r -> infinity is isometric to Euclidean space.
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On $3$-manifolds with small mass and $L^2$-curvature
Affirmative resolution of Yau's problem: small mass implies bilipschitz diffeomorphism to R^3 for 3-manifolds with nonnegative scalar curvature and L^2 curvature bounded by 1.
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Rigidity in the Positive Mass Theorem with $C^0$ Decay
Smooth metrics on R^3 with non-negative scalar curvature and |g - g_euc| = O(|x|^{-1-τ}) for τ>0 are necessarily flat.
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Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions
A Llarull-type rigidity result for scalar curvature holds on odd-dimensional Riemannian spin manifolds with cone-like singularities via twisted Dirac operators and spectral flow.
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On weak formulations of (super) Ricci flows
Smooth compact Ricci flows are characterized weakly solely via metrics and measures by defining super Ricci flows and adding a saturation condition to recover equality.
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