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Existence of complete conformal metrics on $\mathbb{R}^n$ with prescribed Q-curvature

Differential Geometry 2025-04-01 v1 Analysis of PDEs

Abstract

Given a smooth function f(x)f(x) on Rn\mathbb{R}^n which is positive somewhere and satisfies f(x)=O(xl)f(x)=O(|x|^{-l}) for any l>n2l>\frac{n}{2}, we show that there exists a complete and conformal metric g=e2udx2g=e^{2u}|dx|^2 with finite total Q-curvature such that its Q-curvature equals to f(x)f(x).

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Cite

@article{arxiv.2503.23689,
  title  = {Existence of complete conformal metrics on $\mathbb{R}^n$ with prescribed Q-curvature},
  author = {Mingxiang Li and Biao Ma},
  journal= {arXiv preprint arXiv:2503.23689},
  year   = {2025}
}

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14 pages