### Abstract

We prove that conformal metrics on domains of the round sphere, with non-negative constant Q-curvature, and non-negative scalar curvature, has positive mean curvature on the boundary of embedded balls (in the round metric). As a result, such metrics have certain reflection symmetries if the complement of the domain is contained in a lower-dimensional round sphere. We also prove that the development map of a locally conformally flat metric with non-positive Schouten tensor is an embedding.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Birkhauser |

Pages | 37-56 |

Number of pages | 20 |

DOIs | |

State | Published - 2020 |

### Publication series

Name | Progress in Mathematics |
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Volume | 333 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Keywords

- 53C21
- Conformal metrics
- moving planes
- moving spheres
- Primary 58J06
- Q-curvature
- Secondary 35B06
- σk-curvature

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## Cite this

Chang, S. Y. A., Han, Z. C., & Yang, P. (2020). Some Remarks on the Geometry of a Class of Locally Conformally Flat Metrics. In

*Progress in Mathematics*(pp. 37-56). (Progress in Mathematics; Vol. 333). Birkhauser. https://doi.org/10.1007/978-3-030-34953-0_3