A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds
\textit{J. L. Kazdan} and \textit{F. W. Warner} in [Ann. Math. (2) 99, 14--47 (1974; Zbl 0273.53034) and Ann. Math. (2) 99, 203--219 (1974; Zbl 0278.53031)] considered the equation \[ \Delta f+he^f-g=0\tag{*} \] on two-dimensional Riemannian manifolds. In the paper under review, the authors consider this equation on non-compact higher dimensional Riemannian manifolds satisfying the Simpson conditions [\textit{C. T. Simpson}, J. Am. Math. Soc. 1, No.~4, 867--918 (1988; Zbl 0669.58008)], i.e., 1. \(M\) has finite volume, 2. there exists an exhaustion non-negative function \(\varphi\) on \(M\) with \(\Delta\varphi\) bounded, 3. there is an increasing function \(\alpha:[0, \infty)\to [0,\infty)\) with \(\alpha(0)\) and \(\alpha(x)=x\) for \(x>1\), such that if \(f\) is a bounded positive function on \(M\) with \(\Delta f\geq -B\), then \(\sup_M| f| \leq C(B)\alpha\left(\int_M| f| \right).\) Under the above assumptions they prove the following theorem. Let \(h, g\in C^\infty (M)\), \(\sup(| g| +| h|)\leq \infty\). If \(h\leq 0\) is not identically zero and \(\int_Mg<0\), then there is a unique \(f\in C^\infty(M)\) which satisfies the equation (*) and \(\sup_M| f| <\infty\). As an application the authors show existence of Hermitian-Yang-Mills-Higgs metrics on an holomorphic line bundle \(L\) over some non-compact Kähler manifolds \((M, \eta)\).
scientific article; zbMATH DE number 5323382
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