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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.

53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
Full Text: DOI
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