K. K. Uhlenbeck :
“Variational problems for gauge fields ,”
pp. 455–464
in
Seminar on differential geometry .
Edited by S.-T. Yau .
Annals of Mathematics Studies 102 .
Princeton University Press and University of Tokyo Press ,
1982 .
A later article with the same title was published in Proceedings of the International Congress of Mathematicians (1984) .
MR
645753
Zbl
0481.58016
incollection

Abstract
People
BibTeX
@incollection {key645753m,
AUTHOR = {Uhlenbeck, Karen K.},
TITLE = {Variational problems for gauge fields},
BOOKTITLE = {Seminar on differential geometry},
EDITOR = {Yau, Shing-Tung},
SERIES = {Annals of Mathematics Studies},
NUMBER = {102},
PUBLISHER = {Princeton University Press and University
of Tokyo Press},
YEAR = {1982},
PAGES = {455--464},
NOTE = {A later article with the same title
was published in \textit{Proceedings
of the International Congress of Mathematicians}
(1984). MR:645753. Zbl:0481.58016.},
ISSN = {0066-2313},
ISBN = {9781400881918},
}
K. Uhlenbeck and S.-T. Yau :
“On the existence of Hermitian-Yang–Mills connections in stable vector bundles ,”
pp. S257–S293
in
Proceedings of the Symposium on Frontiers of the Mathematical Sciences: 1985
(New York, October 1985 ),
published as Comm. Pure Appl. Math.
39 : Supplement S1 .
Issue edited by C. Morawetz .
J. Wiley and Sons (New York ),
1986 .
A note on this article was published in Commun. Pure Appl. Math. 42 :5 (1989) .
MR
861491
Zbl
0615.58045
incollection

People
BibTeX
@article {key861491m,
AUTHOR = {Uhlenbeck, K. and Yau, S.-T.},
TITLE = {On the existence of {H}ermitian-{Y}ang--{M}ills
connections in stable vector bundles},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {39},
NUMBER = {Supplement S1},
YEAR = {1986},
PAGES = {S257--S293},
DOI = {10.1002/cpa.3160390714},
NOTE = {\textit{Proceedings of the {S}ymposium
on {F}rontiers of the {M}athematical
{S}ciences: 1985} (New York, October
1985). Issue edited by C. Morawetz.
A note on this article was published
in \textit{Commun. Pure Appl. Math.}
\textbf{42}:5 (1989). MR:861491. Zbl:0615.58045.},
ISSN = {0010-3640},
}
K. Uhlenbeck and S. T. Yau :
“A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles ,”
Commun. Pure Appl. Math.
42 : 5
(1989 ),
pp. 703–707 .
A note on an article published in Commun. Pure Appl. Math. 39 :S1 (1986) .
MR
997570
Zbl
0678.58041
article

People
BibTeX
@article {key997570m,
AUTHOR = {Uhlenbeck, K. and Yau, S. T.},
TITLE = {A note on our previous paper: {O}n the
existence of {H}ermitian {Y}ang--{M}ills
connections in stable vector bundles},
JOURNAL = {Commun. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {42},
NUMBER = {5},
YEAR = {1989},
PAGES = {703--707},
DOI = {10.1002/cpa.3160420505},
NOTE = {A note on an article published in \textit{Commun.
Pure Appl. Math.} \textbf{39}:S1 (1986).
MR:997570. Zbl:0678.58041.},
ISSN = {0010-3640},
}
Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck
(Cambridge, MA, 3–5 May 2002 ).
Edited by S.-T. Yau .
Surveys in Differential Geometry 8 .
International Press (Somerville, MA ),
2003 .
Zbl
1034.53003
book

People
BibTeX
@book {key1034.53003z,
TITLE = {Lectures on geometry and topology held
in honor of {C}alabi, {L}awson, {S}iu,
and {U}hlenbeck},
EDITOR = {Yau, Shing-Tung},
SERIES = {Surveys in Differential Geometry},
NUMBER = {8},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {393},
NOTE = {(Cambridge, MA, 3--5 May 2002). Zbl:1034.53003.},
ISSN = {1052-9233},
ISBN = {9781571461148},
}
B. Dai, C.-L. Terng, and K. Uhlenbeck :
“On the space-time monopole equation ,”
pp. 1–30
in
Essays in geometry in memory of S. S. Chern .
Edited by S.-T. Yau .
Surveys in Differential Geometry 10 .
International Press (Somerville, MA ),
2006 .
MR
2408220
Zbl
1157.53016
incollection

Abstract
People
BibTeX

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang–Mills equation on \( \mathbb{R}^{2,2} \) . A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas–Ioannidou, Villarroel and Zakhorov–Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

@incollection {key2408220m,
AUTHOR = {Dai, Bo and Terng, Chuu-Lian and Uhlenbeck,
Karen},
TITLE = {On the space-time monopole equation},
BOOKTITLE = {Essays in geometry in memory of {S}.~{S}.
{C}hern},
EDITOR = {Shing-Tung Yau},
SERIES = {Surveys in Differential Geometry},
NUMBER = {10},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2006},
PAGES = {1--30},
DOI = {10.4310/SDG.2005.v10.n1.a1},
NOTE = {MR:2408220. Zbl:1157.53016.},
ISSN = {1052-9233},
ISBN = {9781571461162},
}
Regularity and evolution of nonlinear equations: Essays dedicated to Richard Hamilton, Leon Simon, and Karen Uhlenbeck .
Edited by H.-D. Cao, R. Schoen, and S.-T. Yau .
Surveys in Differential Geometry 19 .
International Press (Somerville, MA ),
2015 .
MR
3380570
Zbl
1317.53001
book

People
BibTeX
@book {key3380570m,
TITLE = {Regularity and evolution of nonlinear
equations: {E}ssays dedicated to {R}ichard
{H}amilton, {L}eon {S}imon, and {K}aren
{U}hlenbeck},
EDITOR = {Cao, Huai-Dong and Schoen, Richard and
Yau, Shing-Tung},
SERIES = {Surveys in Differential Geometry},
NUMBER = {19},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2015},
PAGES = {vii+301},
NOTE = {MR:3380570. Zbl:1317.53001.},
ISSN = {1052-9233},
ISBN = {9781571463036},
}