J - holomorphic curves and their applications

Wong, Yat Sen

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https://hdl.handle.net/2142/50692 Copy

Description

Title

J - holomorphic curves and their applications

Author(s)

Wong, Yat Sen

Issue Date

2014-09-16

Director of Research (if dissertation) or Advisor (if thesis)

Tumanov, Alexander

Doctoral Committee Chair(s)

Kerman, Ely

Committee Member(s)

• Tumanov, Alexander
• D’Angelo, John
• Tolman, Susan

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Date of Ingest

2014-09-16T17:25:20Z

Keyword(s)

• J-holomorphic curve
• symplectic embedding
• symplectomorphism

Abstract

This thesis covers four results: 1. We prove an analog of Whitney's embedding theorem for J-holomorphic discs. 2. For zj = xj + i*yj in C and let D_R^2 = {(z1, z2) in C^2 : x1^2 + x2^2 <1, y1^2 + y2^2 < 1} be the real bi-disc in C^2. We find the sharp lower bound for R such that D_R^2 admits a symplectic embedding into D(R) * C, the complex cylinder with base radius R. The sharp lower bound for R is shown to be 2/sqrt(pi). As a consequence, we know that D_R^2 and D^2 are not symplectomorphic. 3. We extend the second result by showing that if T is an orthogonal matrix on R^4 = C^2, then TD^2 is symplectomorphic to D^2 if and only if T is unitary or conjugate to unitary. 4. A high dimensional case of the second result: for r >= 1 and n >= 2, we show that D_R^2 * D^(n-2)(r) and D^2 * D^(n-2)(r) are not symplectomorphic.

Graduation Semester

2014-08

Permalink

http://hdl.handle.net/2142/50692

Copyright and License Information

Copyright 2014 Yat Sen Wong

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