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The maximal symmetry rank conjecture

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Let MnM^n be a simply-connected closed manifold with a non-negative curvature metric, and let TkT^k act on MM isometrically and effectively. An action is isotropy maximal if there is a point whose isotropy group has maximal dimension; equivalently, the dimension of a minimal orbit is 2k−n2k-n.

Maximal symmetry rank conjecture. One has

k≤⌊2n3⌋,k\leq \left\lfloor\frac{2n}{3}\right\rfloor,

and the action is isotropy maximal when

k=⌊2n3⌋.k=\left\lfloor\frac{2n}{3}\right\rfloor.

The conjecture describes the expected upper bound for torus symmetry on simply-connected closed non-negatively curved manifolds and the extremal structure of an action attaining that bound. The supplied text does not state its resolution status.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The Maximal Symmetry Rank Conjecture

    Let MnM^n be a closed, simply-connected, non-negatively curved Riemannian manifold with an isometric, effective action of the torus TkT^k. An action is maximal when 2k−n2k-n equals the dimension of the smallest orbit. The Maximal Symmetry Rank Conjecture. If k=⌊2n/3⌋k=\lfloor 2n/3\rfloor, then the action is maximal. This is a reformulation of the maximal symmetry rank problem; the paper proves the conjecture under the additional assumption that the action is almost maximal or maximal, but the unrestricted statement is presented as a conjecture.

    source: Christine Escher and Catherine Searle, “Non-negative curvature and torus actions”, arXiv:1506.08685 (2020).

References

Primary source

Manuel Amann and Leopold Zoller, “The Toral Rank Conjecture and variants of equivariant formality”, arXiv:1910.04746 (2019).

nonnegative-curvaturesymmetry-ranktorus-actions
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