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Conformal metrics in $$\mathbb R^2$$ with prescribed Gaussian curvature and positive total curvature. (English) Zbl 0546.53032
In 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 Hölder 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)
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