Fractional Yamabe solitons and fractional Nirenberg problem

Pak Tung Ho; Rong Tang

doi:10.3934/cpaa.2021103

Article Contents

2021, Volume 20, Issue 9: 3215-3234. Doi: 10.3934/cpaa.2021103

This issue Previous Article A biharmonic transmission problem in L^p-spaces Next Article Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model

Fractional Yamabe solitons and fractional Nirenberg problem

Pak Tung Ho^1,2, , and
Rong Tang^3,

1.
Department of Mathematics, Sogang University, Seoul 04107, Korea, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea
2.
Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea
3.
Department of Mathematics, Johns Hopkins University, Baltimore MD 21218, USA

^* Corresponding author
^* Corresponding author

Received: January 2021

Revised: May 2021

Early access: June 21, 2021

Published online: June 21, 2021

Ho's research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1F1A1041021) and (NRF-2020R1A6A1A03047877), and by Korea Institute for Advanced Study (KIAS) grant funded by the Korea government (MSIP)

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Abstract

In this paper, we first study the fractional Yamabe solitons, which are the self-similar solutions to fractional Yamabe flow.We prove some rigidity results and Liouville type results for such solitons.We thenconsider the fractional Nirenberg problem:the problem of prescribing fractional order curvature on the sphere.More precisely, we prove that there exists a conformal metric on the unit sphere such that itsfractional order curvature is $ f $, when $ f $ possesses certain reflection or rotation symmetry.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 53C21.

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