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The maximal symmetry rank conjecture

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 1

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).

Sources & referencesView supporting material

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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