Conformal metrics in \(\mathbb R^2\) with prescribed Gaussian curvature and positive total curvature.

*(English)*Zbl 0546.53032In recent works of the author and W. M. Ni, it was found that complete Riemannian metrics in \(\mathbb R^2\) can be found which are conformally Euclidean and which have Gaussian curvature \(K(x)\), where \(K(x)\) is a prescribed nonpositive function which decays sufficiently at infinity. In fact, there are infinitely many such metrics, and another quantity which may be prescribed (with a certain range) is the total curvature \(\int_{\mathbb R^2}K(x)\,dA.\) The present paper is concerned with establishing similar results when K(x) is positive or changes sign.

The main result is the following: Theorem. If \(K\) is a HĂ¶lder continuous function on \(\mathbb R^2\) satisfying \(K(x_ 0)>0\) for some \(x_ 0\in \mathbb R^2\) and \(K(x)=O(| x|^{-\ell})\) as \(| x|\to \infty\) where \(\ell >0\), then for every number k satisfying \(\max (0,2\pi (2-\ell))<\kappa <4\pi\) there is a Riemannian metric \(g\) on \(\mathbb R^2\) which is conformally Euclidean, has Gaussian curvature \(K(x)\), and total curvature \(\int_{\mathbb R^2}K(x)\,dA=\kappa.\) Moreover, if \(\ell >1\) then the metric \(4g\) is complete. This result is proved by solving the prescribed curvature equation using variational methods and weighted Sobolev spaces.

The main result is the following: Theorem. If \(K\) is a HĂ¶lder continuous function on \(\mathbb R^2\) satisfying \(K(x_ 0)>0\) for some \(x_ 0\in \mathbb R^2\) and \(K(x)=O(| x|^{-\ell})\) as \(| x|\to \infty\) where \(\ell >0\), then for every number k satisfying \(\max (0,2\pi (2-\ell))<\kappa <4\pi\) there is a Riemannian metric \(g\) on \(\mathbb R^2\) which is conformally Euclidean, has Gaussian curvature \(K(x)\), and total curvature \(\int_{\mathbb R^2}K(x)\,dA=\kappa.\) Moreover, if \(\ell >1\) then the metric \(4g\) is complete. This result is proved by solving the prescribed curvature equation using variational methods and weighted Sobolev spaces.

Reviewer: Robert C. McOwen

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53A30 | Conformal differential geometry (MSC2010) |