Abstract
We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface \(\Sigma \) admitting conical singularities of orders \(\alpha _i\)’s at points \(p_i\)’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity \(\chi (\Sigma )+\sum _i \alpha _i\) approaches a positive even integer, where \(\chi (\Sigma )\) is the Euler characteristic of the surface \(\Sigma \).
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Notes
Here we use the notation \(\sim \) to denote sequences which in the limit \(n\rightarrow +\infty \) are of the same order.
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Acknowledgements
The first author has been supported by the PRIN-Project 201274FYK7_007. The second author has been supported by the PRIN-Project 201274FYK7_005 and by Fondi Avvio alla Ricerca Sapienza 2016. The third author has been supported by the PRIN-Project 201274FYK7_005.
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Communicated by A. Malchiodi.
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D’Aprile, T., De Marchis, F. & Ianni, I. Prescribed Gauss curvature problem on singular surfaces. Calc. Var. 57, 99 (2018). https://doi.org/10.1007/s00526-018-1373-3
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DOI: https://doi.org/10.1007/s00526-018-1373-3