Balanced Hermitian structures on almost abelian Lie algebras |
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| Institution: | 1. Dipartimento di matematica “G. Peano”, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy;2. Department of Mathematics and Statistics, Florida International University, 33199 Miami, USA;1. Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea;2. Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea;1. Department of Mathematics, Rice University, Houston, TX 77005, USA;2. Microsoft Corporation, Redmond, WA 98052, USA;3. Department of Mathematics, The George Washington University, Washington, DC 20052, USA;1. The Ohio State University at Lima, Lima, OH, USA;2. Università di Camerino, Camerino, Italy;1. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Napoli, Italy;2. School of Mathematical Sciences, Queen Mary University of London, London, England, United Kingdom |
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| Abstract: | We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in 1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in 2] and of the anomaly flow 3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points. |
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| Keywords: | Almost abelian Lie algebras Hermitian metrics Balanced metrics Anomaly flow |
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