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Prescribing curvature on compact surfaces with conical singularities. (English) Zbl 0724.53023
The author considers conditions under which a function on a compact Riemann surface is the Gaussian curvature of some Riemannian metric of fixed conformal type with a prescribed set of singularities, all of conical type. This work parallels earlier work by Melvyn Berger and J. Kazdan - F. Warner on prescribing Gaussian curvature functions on a compact Riemann surface without singularities.
The metrics considered here are pointwise conformal, i.e. \(ds^ 2= \rho(x,y)(dx^ 2+dy^ 2),\) where \(z=x+iy\) is a complex parameter on the Riemann surface. A point p of a Riemann surface is a conical singularity of order \(\rho\) (or of angle \(\theta= 2\pi(\rho+1))\) of a metric \(ds^ 2\) if there exists a nonsingular conformal map \(z: U\to {\mathbb{C}}\) defined in a neighborhood U of p such that \(z(p)=0\) and \(ds^ 2=\rho(z) | z|^{2\rho} | dz|^ 2\) in U for some continuous positive function \(\rho\).
The author proves the following results: Theorem A. Let S be a compact Riemann surface. Let \(p_ 1,p_ 2,...,p_ n\) be points of S and \(\theta_ 1,\theta_ 2,...,\theta_ n\) be positive numbers. Assume \(2\pi \chi (S)+\sum^{n}_{i=1}(\theta_ i-2\pi)<0.\) Then any smooth negative function on S is the curvature of a unique conformal metric having at \(p_ i\) a conical singularity of angle \(\theta_ i\). Theorem B. Let S be a compact Riemann surface. Let \(p_ 1,p_ 2,...,p_ n\) be points of S and \(\theta_ 1,\theta_ 2,...,\theta_ n\) be positive numbers. Assume \(2\pi \chi (S)+\sum^{n}_{i=1}(\theta_ i-2\pi)=0.\) Then a smooth function \(K: S\to {\mathbb{R}}\), not identically zero, is the curvature of a conformal metric on S with conical singularities of angle \(\theta_ i\) at \(p_ i\) if and only if i) K changes sign, ii) \(\int_{S}K dA<0\), where dA is the area element of a conformal flat metric having the desired singularities on S. Theorem C. Let S be a compact Riemann surface. Let \(p_ 1,p_ 2,..,p_ n\) be points of S and \(\theta_ 1\leq \theta_ 2\leq...\leq \theta_ n\) be positive numbers. Assume \(0<2\pi\chi(S)+ \sum^{n}_{i=1} (\theta_ i-2\pi)< \min\{4\pi,2\theta_ 1\}.\) Then any smooth function on S which is positive at some point is the curvature of a conformal metric having at \(p_ i\) a conical singularity of angle \(\theta_ i\). The results of this article extend to nonorientable surfaces and to surfaces with piecewise geodesic boundary.

MSC:
53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
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