Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space
type: Item
instance of: scholarly article
title: Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space
zbMATH Open document ID: 1485.53045
author: Natsuo Miyatake
published in: Geometriae Dedicata
publication date: 2021-09-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/2002.09141
review text: 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.
reviewed by: Garik Petrosyan
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35R01
Mathematics Subject Classification ID: 53D20
zbMATH DE Number: 7402701
zbMATH Keywords: Kazdan-Warner equation
zbMATH Keywords: vortex equation
zbMATH Keywords: cyclic Higgs bundles
zbMATH Keywords: periodic Toda equation
zbMATH Keywords: moment map
zbMATH Keywords: geometric invariant theory
zbMATH Keywords: Higgs bundles
MaRDI profile type: Publication
OpenAlex ID: W3168779069
arXiv ID: 2002.09141
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DOI: 10.1007/S10711-021-00632-Z
description: scientific article