Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 1

Special Issue in Honor of Bernie Shiffman

Guest Editors: Yuan Yuan, Christopher Sogge, and Steven Morris Zelditch

The exponential map of the complexification of the group of analytic Hamiltonian diffeomorphisms

Pages: 33 – 70

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n1.a2

Authors

Reebhu Bhattacharyya (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Dan Burns (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Ernesto Lupercio (Departamento de Matemáticas, CINVESTAV-IPN, Ciudad de México, México)

Alejandro Uribe (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Abstract

Let $(M,\omega,J)$ be a Kähler manifold and $K = \operatorname{Ham}(M,\omega)$ its group of Hamiltonian symplectomorphisms. Complexifications of $K$ have been introduced by Semmes and then Donaldson which are not groups, only “formal Lie group” in a precise sense. However, it still makes sense to talk about the exponential map in the complexification. In this note we give a geometric construction of the exponential map (for small time), in case the initial data are real-analytic. (A more general analytic description has been given by Semmes.) The construction is motivated by, but does not use, semiclassical analysis and quantum coherent states. We use this geometric construction to solve the equations of motion in several basic examples and recapture a case already considered in the physics community where the quantum analogue of our system is explicitly solvable, showing a potential relation to non-Hermitian quantum mechanics. Finally, in the case of geodesics in the space of Kähler metrics on a Kähler manifold originally studied variously by Mabuchi, Semmes and Donaldson, we derive an infinitesimal obstruction to the completeness of Mabuchi geodesic rays in the space of smooth Kähler metrics.

2010 Mathematics Subject Classification

Primary 53C55, 58D05. Secondary 81S10.

Received 13 July 2020

Received revised 27 August 2021

Accepted 13 September 2021

Published 10 February 2022