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The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of F4(−20)

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Abstract

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positive OC manifolds to be scalar positive if the neck is sufficiently long. On the OC manifold constructed from a convex cocompact subgroup of F4(−20), we construct a Nayatani-type Carnot–Carathéodory metric. As a corollary, such an OC manifold is scalar positive, negative or vanishing if and only if the Poincaré critical exponent of the subgroup is less than, greater than or equal to 10,  respectively.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Yun Shi

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Wei Wang

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  1. Yun Shi
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  2. Wei Wang
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Correspondence to Wei Wang.

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The first author is partially supported by National Nature Science Foundation in China (Nos. 11801508, 11971425); The second author is partially supported by National Nature Science Foundation in China (No. 11971425).

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Shi, Y., Wang, W. The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of F4(−20). Annali di Matematica 200, 2597–2630 (2021). https://doi.org/10.1007/s10231-021-01093-7

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  • DOI: https://doi.org/10.1007/s10231-021-01093-7

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