Toric, U(2), and LeBrun metrics

Abstract The LeBrun ansatz was designed for scalar-flat Kähler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which i...

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Autor principal:	Weber,Brian
Lenguaje:	English
Publicado:	Universidad de La Frontera. Departamento de Matemática y Estadística. 2020
Materias:	Differential geometry Kähler geometry canonical metrics ansatz.
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462020000300395
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spelling	oai:scielo:S0719-064620200003003952020-12-30Toric, U(2), and LeBrun metricsWeber,Brian Differential geometry Kähler geometry canonical metrics ansatz. Abstract The LeBrun ansatz was designed for scalar-flat Kähler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is Kähler, scalar-flat, or extremal Kähler. Second, toric Kähler metrics (such as the generalized Taub-NUTs) and U(2)-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the LeBrun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples-the exceptional half-plane metric and the Page metric-in terms of the LeBrun ansatz.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.22 n.3 20202020-12-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462020000300395en10.4067/S0719-06462020000300395
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Differential geometry Kähler geometry canonical metrics ansatz.
spellingShingle	Differential geometry Kähler geometry canonical metrics ansatz. Weber,Brian Toric, U(2), and LeBrun metrics
description	Abstract The LeBrun ansatz was designed for scalar-flat Kähler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is Kähler, scalar-flat, or extremal Kähler. Second, toric Kähler metrics (such as the generalized Taub-NUTs) and U(2)-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the LeBrun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples-the exceptional half-plane metric and the Page metric-in terms of the LeBrun ansatz.
author	Weber,Brian
author_facet	Weber,Brian
author_sort	Weber,Brian
title	Toric, U(2), and LeBrun metrics
title_short	Toric, U(2), and LeBrun metrics
title_full	Toric, U(2), and LeBrun metrics
title_fullStr	Toric, U(2), and LeBrun metrics
title_full_unstemmed	Toric, U(2), and LeBrun metrics
title_sort	toric, u(2), and lebrun metrics
publisher	Universidad de La Frontera. Departamento de Matemática y Estadística.
publishDate	2020
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462020000300395
work_keys_str_mv	AT weberbrian toricu2andlebrunmetrics
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Toric, U(2), and LeBrun metrics

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