Scalar Curvature, Convex Polyhedra & Differential Operators

Prof Misha Gromov gave two lectures at Stanford on Nov 10 and Nov 11. Here is a brief summary of his Nov 10 lecture:
Scalar curvature influences topology and asymptotic geometry of Riemannian manifolds, it is associated with the Einstein equation, it controls the Hamilton-Ricci flow. Yet, the geometric meaning of the scalar curvature remains obscure.
We shall explain in this lecture what can be seen of the scalar curvature from the
perspective suggested by geometry and combinatorics of convex polyhedra.
  • What polyhedrons have the property that all dihedral angles are less than or equal to \frac{\pi}{2}?
  • What about strictly less than \frac{\pi}{2}?

The second question is no for positive scalar curvature spaces.

  • Dirac operator on Riemannian spin manifolds
  • R. Schoen + S.T. Yau: can’t give torus of dimension 3 positive scalar curvature.

Extension: same applies for dimension 4,5,6,7. Fails at 8 and above. Proof uses Gauss formula.

  • Can’t deform a manifold to n sphere if the scalar curvature is bigger than that of the sphere.

Proof using Yau’s theorem which states we can’t deform so that polyhedral angles go down while still preserving the scalar curvature.

  • Positive scalar curvature \epsilon-balls volumewise sub-Euclidean.

 

It seems Gromov went over the more crucial and understandable results from papers on his homepage, section 8.So you can find detailed descriptions as well as proofs for those bullet points under one of his papers.

Later I found this by Shing-Tung Yao as well: Topics on Geometric Analysis. Not directly related to this talk, but still a fun read!

 

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