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Bi-invariant norms on the group of s...
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State University of New York at Stony Brook.
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Bi-invariant norms on the group of symplectomorphisms.
Record Type:
Language materials, printed : Monograph/item
Title/Author:
Bi-invariant norms on the group of symplectomorphisms./
Author:
Han, Zhigang.
Description:
64 p.
Notes:
Adviser: Dusa McDuff.
Contained By:
Dissertation Abstracts International68-01B.
Subject:
Mathematics. -
Online resource:
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3246833
Bi-invariant norms on the group of symplectomorphisms.
Han, Zhigang.
Bi-invariant norms on the group of symplectomorphisms.
- 64 p.
Adviser: Dusa McDuff.
Thesis (Ph.D.)--State University of New York at Stony Brook, 2006.
The group of diffeomorphisms of a symplectic manifold (M, o) preserving the symplectic form is called the symplectomorphism group of (M, o) and denoted by Symp(M, o). It has a very important subgroup Ham( M, o) called the Hamiltonian diffeomorphism group. The group Ham(M, o) admits a natural bi-invariant norm, known as the Hofer norm. In this thesis we study several aspects of bi-invariant norms on Symp(M, o), including the bounded isometry conjecture of Lalonde and Polterovich. In particular, we prove the conjecture for the Kodaira-Thurston manifold and for the 4-torus with all linear symplectic forms. Relatedly, we observe that there is an obstruction to extending the Hofer norm bi-invariantly to the identity component Symp0( M, o) of Symp(M, o). This obstruction is shown to be non-trivial in some cases. We also prove that no Finsler norm on Ham( T2n , o) satisfying a strong form of the invariance condition can extend to a bi-invariant norm on Symp0( T2n , o). Other bi-invariant norms on Symp0(M, o) are studied as well and the induced topologies are discussed.Subjects--Topical Terms:
515831
Mathematics.
Bi-invariant norms on the group of symplectomorphisms.
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Bi-invariant norms on the group of symplectomorphisms.
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64 p.
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Adviser: Dusa McDuff.
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Source: Dissertation Abstracts International, Volume: 68-01, Section: B, page: 0326.
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Thesis (Ph.D.)--State University of New York at Stony Brook, 2006.
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The group of diffeomorphisms of a symplectic manifold (M, o) preserving the symplectic form is called the symplectomorphism group of (M, o) and denoted by Symp(M, o). It has a very important subgroup Ham( M, o) called the Hamiltonian diffeomorphism group. The group Ham(M, o) admits a natural bi-invariant norm, known as the Hofer norm. In this thesis we study several aspects of bi-invariant norms on Symp(M, o), including the bounded isometry conjecture of Lalonde and Polterovich. In particular, we prove the conjecture for the Kodaira-Thurston manifold and for the 4-torus with all linear symplectic forms. Relatedly, we observe that there is an obstruction to extending the Hofer norm bi-invariantly to the identity component Symp0( M, o) of Symp(M, o). This obstruction is shown to be non-trivial in some cases. We also prove that no Finsler norm on Ham( T2n , o) satisfying a strong form of the invariance condition can extend to a bi-invariant norm on Symp0( T2n , o). Other bi-invariant norms on Symp0(M, o) are studied as well and the induced topologies are discussed.
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http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3246833
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