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Torus actions, maximality, and non-negative curvature
Escher, C., & Searle, C. (2021). Torus actions, maximality, and non-negative curvature. Journal Fur Die Reine Und Angewandte Mathematik, doi:10.1515/crelle-2021-0035
Let $ℳ_0^n$ be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M \in ℳ_0^n$ then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all $M \in ℳ_0^n$. Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.
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