Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

We show for a certain class of operators A and holomorphic functions f that the functional calculus A↦ f(A) is holomorphic. Using this result we are able to prove that fractional Laplacians (1+Δg)p depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain l...

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Main Authors:	Bauer, Martin, Bruveris, Martins, Harms, Philipp, Michor, Peter W.
Other Authors:	School of Physical and Mathematical Sciences
Format:	Article
Language:	English
Published:	2022
Subjects:	Science::Mathematics Riemannian Geometry Holomorphic Functions
Online Access:	https://hdl.handle.net/10356/162510
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Institution:	Nanyang Technological University
Language:	English

id	sg-ntu-dr.10356-162510
record_format	dspace
spelling	sg-ntu-dr.10356-1625102022-10-26T02:56:02Z Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics Bauer, Martin Bruveris, Martins Harms, Philipp Michor, Peter W. School of Physical and Mathematical Sciences Science::Mathematics Riemannian Geometry Holomorphic Functions We show for a certain class of operators A and holomorphic functions f that the functional calculus A↦ f(A) is holomorphic. Using this result we are able to prove that fractional Laplacians (1+Δg)p depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics. MB was partially supported by NSF-grants 1912037 and 1953244. PH was partially supported in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies. 2022-10-26T02:56:01Z 2022-10-26T02:56:01Z 2022 Journal Article Bauer, M., Bruveris, M., Harms, P. & Michor, P. W. (2022). Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics. Communications in Mathematical Physics, 389(2), 899-931. https://dx.doi.org/10.1007/s00220-021-04264-y 0010-3616 https://hdl.handle.net/10356/162510 10.1007/s00220-021-04264-y 2-s2.0-85122655630 2 389 899 931 en Communications in Mathematical Physics © The Author(s) 2021 under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
institution	Nanyang Technological University
building	NTU Library
continent	Asia
country	Singapore Singapore
content_provider	NTU Library
collection	DR-NTU
language	English
topic	Science::Mathematics Riemannian Geometry Holomorphic Functions
spellingShingle	Science::Mathematics Riemannian Geometry Holomorphic Functions Bauer, Martin Bruveris, Martins Harms, Philipp Michor, Peter W. Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
description	We show for a certain class of operators A and holomorphic functions f that the functional calculus A↦ f(A) is holomorphic. Using this result we are able to prove that fractional Laplacians (1+Δg)p depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.
author2	School of Physical and Mathematical Sciences
author_facet	School of Physical and Mathematical Sciences Bauer, Martin Bruveris, Martins Harms, Philipp Michor, Peter W.
format	Article
author	Bauer, Martin Bruveris, Martins Harms, Philipp Michor, Peter W.
author_sort	Bauer, Martin
title	Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
title_short	Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
title_full	Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
title_fullStr	Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
title_full_unstemmed	Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
title_sort	smooth perturbations of the functional calculus and applications to riemannian geometry on spaces of metrics
publishDate	2022
url	https://hdl.handle.net/10356/162510
_version_	1749179137633812480

Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

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