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Existence results for the prescribed Gauss curvature problem on closed surfaces D'Aprile, Teresa


Let $(\Sigma, g)$ be a compact orientable surface without boundary and with metric $g$ and Gauss curvature $\kappa_g$. Given points $p_i\in \Sigma$ and a Lipschitz function $K$ defined on $\Sigma$, a classical problem in differential geometry is the question on the existence of a metric $\tilde g$ conformal to $g$ in $\Sigma\setminus\{p_1,\ldots, p_m\}$, namely $$\tilde g = e^{u} g\hbox{ in }\Sigma\setminus\{p_1,\ldots, p_m\}$$ admitting conical singularities of orders $\alpha_i$'s (with $\alpha_i>-1$) at the points $p_i$'s and having $K$ as the associated Gaussian curvature in $\Sigma\setminus\{p_1,\ldots, p_m\}$. The question reduces to solving a singular Liouville-type equation on $\Sigma$ \begin{equation*}-\Delta_g u+2\kappa_g =2K e^u-4\pi \sum_{i=1}^m\alpha_i\delta_{p_i}\hbox{ in }\Sigma\end{equation*} where $\delta_p$ denotes Dirac mass supported at $p$. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new existence results in the perturbative regime when the quantity $\chi(\Sigma)+\sum_{i=1}^m \alpha_i$ approaches a positive even integer, where $\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$.

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