Please use this identifier to cite or link to this item:

`http://arks.princeton.edu/ark:/88435/dsp01sf2685239`

Title: | Some Results on a Fully Nonlinear Equation in Conformal Geometry |

Authors: | Reichert, Nicholas William |

Advisors: | Chang, Sun-Yung Alice Yang, Paul Chien-Ping |

Contributors: | Mathematics Department |

Keywords: | Conformal Geometry Differential Geometry Partial Differential Equations |

Subjects: | Mathematics |

Issue Date: | 2014 |

Publisher: | Princeton, NJ : Princeton University |

Abstract: | In this thesis we study the question of solving an equation arising in conformal geometry. Specifically, we obtain information about the equation σk<\sub>(g<super>−1<\super>Ag<\sub>) = constant. Here g is a metric in a given conformal class [g0<\sub>] and Ag is the Schouten tensor of g. First, we generalize a result of Chang, Gursky, and Yang to show that under certain geometric conditions, entire solutions of this equation on R<super>n</super> must necessarily arise via pullback from the sphere under stereographic projection. This “Obata” type theorem provides an alternate proof of a result of Li and Li. However, our approach uses integral estimates which may be more amenable to application on manifolds which are not locally conformally flat. Second, we study the question of solving σk<\sub> = constant on n-manifolds M, the “k-Yamabe problem“, when k = 2 and n = 3. The k-Yamabe problem has been studied for many values of n and k before (by Li and Li, by Guan and Wang, by Gursky and Viaclovsky, by Sheng, Trudinger, and Wang, and by Trudinger and Wang, among others); however, some cases remain open. The case where n < 2k is unresolved in general. A perhaps more natural question to consider in this case is solving v2k<\sub> = constant, for n < 2k, where v2k<\sub> are the renormalized volume coefficients studied by Graham and by Chang and Fang. Unfortunately, the approach used for other values of n and k does not apply in this final case-for v2k<\sub>, one cannot use the local estimates of Guan and Wang and of Chen. This is because the local estimates use algebraic properties of k which do not hold for v2k<\sub>. Though the σ2<\sub>-Yamabe problem in dimension 3 has been studied before, we study an alternative approach which avoids the use of the local estimates. We hope this approach may prove useful for resolving the question of solving v2k<\sub> = constant. |

URI: | http://arks.princeton.edu/ark:/88435/dsp01sf2685239 |

Alternate format: | The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog |

Type of Material: | Academic dissertations (Ph.D.) |

Language: | en |

Appears in Collections: | Mathematics |

Files in This Item:

File | Description | Size | Format | |
---|---|---|---|---|

Reichert_princeton_0181D_10852.pdf | 480.44 kB | Adobe PDF | View/Download |

Items in Dataspace are protected by copyright, with all rights reserved, unless otherwise indicated.