Boundedness of Differential of Symplectic Vortices in Open Cylinder Model
Let <i>G</i> be a compact connected Lie group, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><m...
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| author | Hai-Long Her |
| author_facet | Hai-Long Her |
| author_sort | Hai-Long Her |
| collection | DOAJ |
| description | Let <i>G</i> be a compact connected Lie group, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> a Hamiltonian <i>G</i>-manifold with moment map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, and <i>Z</i> a codimension-2 Hamiltonian <i>G</i>-submanifold of <i>X</i>. We study the boundedness of the differential of symplectic vortices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> near <i>Z</i>, where <i>A</i> is a connection 1-form of a principal <i>G</i>-bundle <i>P</i> over a punctured Riemann surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mo>Σ</mo><mo>˚</mo></mover></semantics></math></inline-formula>, and <i>u</i> is a <i>G</i>-equivariant map from <i>P</i> to an open cylinder model near <i>Z</i>. We show that if the total energy of a family of symplectic vortices on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mo>Σ</mo><mo>˚</mo></mover><mo>≅</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></mrow></semantics></math></inline-formula> is finite, then the <i>A</i>-twisted differential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>A</mi></msub><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is uniformly bounded for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-b30e857785a7401491302bfc39f9f793 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-b30e857785a7401491302bfc39f9f7932025-08-20T02:05:04ZengMDPI AGMathematics2227-73902024-11-011222349810.3390/math12223498Boundedness of Differential of Symplectic Vortices in Open Cylinder ModelHai-Long Her0Department of Mathematics, Jinan University, Guangzhou 510632, ChinaLet <i>G</i> be a compact connected Lie group, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> a Hamiltonian <i>G</i>-manifold with moment map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, and <i>Z</i> a codimension-2 Hamiltonian <i>G</i>-submanifold of <i>X</i>. We study the boundedness of the differential of symplectic vortices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> near <i>Z</i>, where <i>A</i> is a connection 1-form of a principal <i>G</i>-bundle <i>P</i> over a punctured Riemann surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mo>Σ</mo><mo>˚</mo></mover></semantics></math></inline-formula>, and <i>u</i> is a <i>G</i>-equivariant map from <i>P</i> to an open cylinder model near <i>Z</i>. We show that if the total energy of a family of symplectic vortices on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mo>Σ</mo><mo>˚</mo></mover><mo>≅</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></mrow></semantics></math></inline-formula> is finite, then the <i>A</i>-twisted differential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>A</mi></msub><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is uniformly bounded for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/12/22/3498symplectic manifoldHamiltonian actionmoment mapsymplectic vortex |
| spellingShingle | Hai-Long Her Boundedness of Differential of Symplectic Vortices in Open Cylinder Model Mathematics symplectic manifold Hamiltonian action moment map symplectic vortex |
| title | Boundedness of Differential of Symplectic Vortices in Open Cylinder Model |
| title_full | Boundedness of Differential of Symplectic Vortices in Open Cylinder Model |
| title_fullStr | Boundedness of Differential of Symplectic Vortices in Open Cylinder Model |
| title_full_unstemmed | Boundedness of Differential of Symplectic Vortices in Open Cylinder Model |
| title_short | Boundedness of Differential of Symplectic Vortices in Open Cylinder Model |
| title_sort | boundedness of differential of symplectic vortices in open cylinder model |
| topic | symplectic manifold Hamiltonian action moment map symplectic vortex |
| url | https://www.mdpi.com/2227-7390/12/22/3498 |
| work_keys_str_mv | AT hailongher boundednessofdifferentialofsymplecticvorticesinopencylindermodel |