Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry
Nyström, David Witt
International Mathematics Research Notices
Oxford University Press
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Ross, J., & Nyström, D. W. (2018). Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry. International Mathematics Research Notices, 2017 (23), 7069-7108. https://doi.org/10.1093/imrn/rnw200
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Amp$\grave e$re equation on the deformation to the normal cone of the submanifold. We use this to establish local regularity for global weak solutions, giving local smoothness to the (weak) geodesic ray in the space of (weak) Kähler potentials associated to a given complex submanifold. We also use it to get an optimal regularity result for naturally defined plurisubharmonic envelopes and for the boundaries of their associated equilibrium sets.
During this work JR was supported by an EPSRC Career Acceleration Fellowship (EP/J002062/1). DWN has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no 329070, and previously by Chalmers University and the University of Gothenburg.
External DOI: https://doi.org/10.1093/imrn/rnw200
This record's URL: https://www.repository.cam.ac.uk/handle/1810/260896