Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space
In this paper the authors introduce generalized Kazdan-Warner equations on Riemannian manifolds \((M,g_M)\) of the following form: \[ \Delta_{g_{M}}\xi+\sum^{d}_{j=1}a_j e^{(i^* u^j ,\xi)}i^* u^j = w, \eqno (1) \] where for a closed connected subtorus \(K\) of a real torus \(T^d:=U(1)^d\) with the Lie algebra \(k\subseteq t^d\) with the metric \((\cdot,\cdot),\) \(u^j, j=1,...,d,\) the dual basis of the special basis \(u_i, i=1,...,d,\) in \(t^d,\) \(i^* :(t^d)^* \to k^*\) is the dual map of the inclusion map \(i:k\to t^d.\) A function \(\xi\) is a \(k^*\)-valued function on \(M\) which is the solution of (1) for given nonnegative functions \(a_j, j=1,...,d,\) \(w\) is a \(k^*\)-valued function and \(\Delta_{g_{M}}\) is the geometric Laplacian. The authors proof the existence and the uniqueness of the solution of equation (1) on any compact Riemannian manifold and as an application, they give a new proof of a theorem of Baraglia which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation.
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