Calculus of Variations and Geometric Measure Theory

C. Bernardini

Existence and Compactness of Conformal Metrics on the Plane with Unbounded and Sign-Changing Gaussian Curvature

created by bernardini on 15 Dec 2022


Accepted Paper

Inserted: 15 dec 2022

Journal: Vietnam J. Math.
Year: 2021
Doi: 10.1007/s10013-021-00540-5


We show that the prescribed Gaussian curvature equation in $\mathbb{R}^2$ \[-\Delta u= (1-
^p) e^{2u},\] has solutions with prescribed total curvature equal to $\Lambda:=\int_{\mathbb{R}^2}(1-
^p)e^{2u}dx\in \mathbb{R}$, if and only if \[p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le\Lambda<4\pi\] and prove that such solutions remain compact as $\Lambda\to\bar{\Lambda}\in[(2+p)\pi,4\pi)$, while they produce a spherical blow-up as $\Lambda\uparrow4\pi$.\\