Abstract
We introduce a flow approach to the generalized Loewner-Nirenberg problem (1.5)-(1.7) of the \(\sigma _k\)-Ricci equation on a compact manifold \((M^n,g)\) with boundary. We prove that for initial data \(u_0\in C^{4,\alpha }(M)\) which is a subsolution to the \(\sigma _k\)-Ricci equation (1.5), the Cauchy-Dirichlet problem (3.1)-(3.3) has a unique solution u which converges in \(C^4_{loc}(M^{\circ })\) to the solution \(u_{\infty }\) of the problem (1.5)-(1.7), as \(t\rightarrow \infty \).
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Acknowledgements
The author would like to thank Professor Matthew Gursky for helpful discussion and Professor Jiakun Liu for nice talks on nonlinear equations.
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Communicated by S. A. Chang.
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G. Li: Research partially supported by the National Natural Science Foundation of China No. 11701326 and the Young Scholars Program of Shandong University 2018WLJH85.
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Li, G. A flow approach to the generalized Loewner-Nirenberg problem of the \(\sigma _k\)-Ricci equation. Calc. Var. 61, 169 (2022). https://doi.org/10.1007/s00526-022-02283-8
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DOI: https://doi.org/10.1007/s00526-022-02283-8