Sur les difféomorphismes de la sphère de dimension trois (Г[4]=0)

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

Four Lectures on Scalar Curvature

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (
 $$L^\infty $$
 ) metrics that consolidate Gromov’s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles $$\le 2\pi $$
 along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles $$\le 2\pi $$
 ) and arbitrary $$L^\infty $$
 isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.

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Positive scalar curvature with skeleton singularities

A metric space $X$ is called uniformly acyclic if there there exists an {\it acyclicty control function} $R=R(r)=R_X(r)\geq r $, $0\leq r 0$ for $dim (X)\leq 5$. Our argument, that depends on {\it torical symmetrization} of {\it stable $\mu$-bubbles}, is inspired by the recent paper by Otis Chodosh and Chao Li on non-existence of metrics with $Sc>0$ on aspherical 4-manifolds and is also influenced by the ideas of Jintian Zhu and Thomas Richard.

No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds

We prove that for $n\in \{4,5\}$, a closed aspherical $n$-manifold does not admit a Riemannian metric with positive scalar curvature. 
Additionally, we show that for $n\leq 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with forthcoming contributions by Lesourd--Unger--Yau, this proves that the Schoen--Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature. 
A key tool in these results are generalized soap bubbles---surfaces that are stationary for prescribed-mean-curvature functionals (also called $\mu$-bubbles).

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

Constrained deformations of positive scalar curvature metrics, II

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

Chao Li

Papers

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

Positive scalar curvature with skeleton singularities

Constrained deformations of positive scalar curvature metrics, II

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature