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Gradient ambient obstruction solitons on homogeneous manifolds

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Abstract

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on \(\mathbb {R}^2\times S^2\) and \(\mathbb {R}^2 \times H^2\). We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor q.

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Acknowledgements

The author would like to thank Professor Dylan Helliwell of Seattle University and Professor Peter Petersen of UCLA for their interest and helpful discussions when writing this paper. Thank you also to my thesis advisor, Professor William Wylie of Syracuse University, for his guidance, support, and insight into this topic.

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  1. Syracuse University, Syracuse, New York, USA

    Erin Griffin

  2. Seattle Pacific University, Seattle, Washington, USA

    Erin Griffin

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  1. Erin Griffin
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Correspondence to Erin Griffin.

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Griffin, E. Gradient ambient obstruction solitons on homogeneous manifolds. Ann Glob Anal Geom 60, 469–499 (2021). https://doi.org/10.1007/s10455-021-09784-3

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  • DOI: https://doi.org/10.1007/s10455-021-09784-3

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