A dual Yamabe flow and related integral flows
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910289701634048 |
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| author | Xiong, Jingang |
| author_facet | Xiong, Jingang |
| contents | We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15852 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A dual Yamabe flow and related integral flows Xiong, Jingang Analysis of PDEs Classical Analysis and ODEs Differential Geometry We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds. |
| title | A dual Yamabe flow and related integral flows |
| topic | Analysis of PDEs Classical Analysis and ODEs Differential Geometry |
| url | https://arxiv.org/abs/2312.15852 |