We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface Σ admitting conical singularities of orders αi ’s at points pi ’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 χ(Σ)+∑iαi approaches a positive even integer, where χ(Σ) is the Euler characteristic of the surface Σ .
D'Aprile, T., De Marchis, F., Ianni, I. (2018). Prescribed Gauss curvature problem on singular surfaces. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 57(4) [10.1007/s00526-018-1373-3].
Prescribed Gauss curvature problem on singular surfaces
D'Aprile T.;
2018-08-01
Abstract
We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface Σ admitting conical singularities of orders αi ’s at points pi ’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 χ(Σ)+∑iαi approaches a positive even integer, where χ(Σ) is the Euler characteristic of the surface Σ .File | Dimensione | Formato | |
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