Abstract
Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators \(C(S)\), which are nonnegative in a suitable sense, to every \(Ad_{SO(n,\mathbb{C })}\) invariant subset \(S \subset \mathbf{so}(n,\mathbb{C })\). In this article we show that if \(S\) is an \(Ad_{SO(n,\mathbb{C })}\) invariant subset of \(\mathbf{so}(n,\mathbb{C })\) such that \(S\cup \{0\}\) is closed and \(C_+(S)\subset C(S)\) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in \(C_+(S)\) also admits a metric with curvature operator in \(C_+(S)\) (b) The normalized Ricci flow on any compact Riemannian manifold \(M\) with curvature operator in \(C_+(S)\) converges to a metric of constant positive sectional curvature. We also point out that if \(S\) is an arbitrary \(Ad_{SO(n,\mathbb{C })}\) subset, then \(C(S)\) is contained in the cone of curvature operators with nonnegative isotropic curvature.
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Gururaja, H.A., Maity, S. & Seshadri, H. On Wilking’s criterion for the Ricci flow. Math. Z. 274, 471–481 (2013). https://doi.org/10.1007/s00209-012-1079-8
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DOI: https://doi.org/10.1007/s00209-012-1079-8