2026-09-19T07:07:20Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1046072026-06-16T18:15:07Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Tomasz Mrowka.
Berchenko-Kogan, Yakov
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2016-09-30T19:38:05Z
2016-09-30T19:38:05Z
2016
2016
http://hdl.handle.net/1721.1/104607
958973139
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 87-88).
We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function v: ... defined on a surface ... and replacing its values on a small ball B2 ... with a harmonic function u that has the same values as v on the boundary &B2 . The resulting function on ... has lower energy, and repeating this process on balls covering ..., one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection B on a bundle over a four-manifold X, and replace it on a small ball ... with a Yang-Mills connection A that has the same restriction to the boundary [alpha]B4 as B, and we obtain bounds on the difference ... in terms of the drop in energy. Throughout, we work with connections of the lowest possible regularity ... (X), the natural choice for this context, and so our gauge transformations are in ... (X) and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.
by Yakov Berchenko-Kogan.
Ph.D.
88 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Yang-Mills replacement
Thesis
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<Abstract>We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function v: ... defined on a surface ... and replacing its values on a small ball B2 ... with a harmonic function u that has the same values as v on the boundary &B2 . The resulting function on ... has lower energy, and repeating this process on balls covering ..., one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context. We take a connection B on a bundle over a four-manifold X, and replace it on a small ball ... with a Yang-Mills connection A that has the same restriction to the boundary [alpha]B4 as B, and we obtain bounds on the difference ... in terms of the drop in energy. Throughout, we work with connections of the lowest possible regularity ... (X), the natural choice for this context, and so our gauge transformations are in ... (X) and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.</Abstract>
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