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arXiv · 1612.03657

Prescribed Gauss curvature problem on singular surfaces

Teresa D'Aprile·Francesca De Marchis·Isabella Ianni
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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 $α_i$'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 $χ(Σ)+\sum_i α_i$ approaches a positive even integer, where $χ(Σ)$ is the Euler characteristic of the surface $Σ$.

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Teresa D'Aprile, Francesca De Marchis, Isabella Ianni. 2017-01-19. Prescribed Gauss curvature problem on singular surfaces. https://arxiv.org/abs/1612.03657

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