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Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions
About this Title
Paul M. N. Feehan and Manousos Maridakis
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 267, Number 1302
ISBNs: 978-1-4704-4302-3 (print); 978-1-4704-6403-5 (online)
DOI: https://doi.org/10.1090/memo/1302
Published electronically: January 12, 2021
Keywords: Gauge theory,
Łojasiewicz–Simon gradient inequality,
Morse–Bott theory on Banach manifolds,
smooth four-dimensional manifolds,
coupled Yang–Mills equations
Table of Contents
Chapters
- Preface
- 1. Introduction
- 2. Existence of Coulomb gauge transformations for connections and pairs
- 3. Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions
- 4. Łojasiewicz–Simon $W^{-1,2}$ gradient inequalities for coupled Yang–Mills energy functions
- A. Fredholm and index properties of elliptic operators on Sobolev spaces
- B. Equivalence of Sobolev norms defined by Sobolev and smooth connections
- C. Fredholm and index properties of a Hodge Laplacian with Sobolev coefficients
- D. Convergence of gradient flows under the validity of the Łojasiewicz–Simon gradient inequality
- E. Huang’s Łojasiewcz–Simon gradient inequality for analytic functions on Banach spaces
- F. Quantitative implicit and inverse function theorems
Abstract
We prove Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions generalize that of the pure Yang–Mills energy function due to the first author (Feehan, 2014) for base manifolds of arbitrary dimension and due to Råde (1992, Proposition 7.2) for dimensions two and three.- R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687, DOI 10.1007/978-1-4612-1029-0
- Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782, DOI 10.1090/gsm/050
- A. G. Ache, On the uniqueness of asymptotic limits of the Ricci flow, arXiv:1211.3387.
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Tom M. Apostol, Mathematical analysis, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974. MR 344384
- Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
- Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 488101
- Jean-Pierre Bourguignon, Formules de Weitzenböck en dimension $4$, Riemannian geometry in dimension 4 (Paris, 1978/1979) Textes Math., vol. 3, CEDIC, Paris, 1981, pp. 308–333 (French). MR 769143
- Jean-Pierre Bourguignon, The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 251–271. MR 1205158
- Steven B. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1990), no. 1, 1–17. MR 1086749
- Steven B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Differential Geom. 33 (1991), no. 1, 169–213. MR 1085139
- Steven B. Bradlow and Oscar García-Prada, Non-abelian monopoles and vortices, Geometry and physics (Aarhus, 1995) Lecture Notes in Pure and Appl. Math., vol. 184, Dekker, New York, 1997, pp. 567–589. MR 1423193, DOI 10.2307/2500944
- Simon Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. 69 (2005), no. 2, 217–278. MR 2168505
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR 1410059
- Alessandro Carlotto, Otis Chodosh, and Yanir A. Rubinstein, Slowly converging Yamabe flows, Geom. Topol. 19 (2015), no. 3, 1523–1568. MR 3352243, DOI 10.2140/gt.2015.19.1523
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Ralph Chill, On the Łojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), no. 2, 572–601. MR 1986700, DOI 10.1016/S0022-1236(02)00102-7
- R. Chill, The Łojasiewicz–Simon gradient inequality in Hilbert spaces, Proceedings of the 5th European-Maghrebian Workshop on Semigroup Theory, Evolution Equations, and Applications (M. A. Jendoubi, ed.), 2006, pp. 25–36.
- Ralph Chill and Alberto Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Differential Equations 228 (2006), no. 2, 611–632. MR 2289546, DOI 10.1016/j.jde.2006.02.009
- Ralph Chill, Alain Haraux, and Mohamed Ali Jendoubi, Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. Appl. (Singap.) 7 (2009), no. 4, 351–372. MR 2572850, DOI 10.1142/S0219530509001438
- R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. 53 (2003), no. 7-8, 1017–1039. MR 1978032, DOI 10.1016/S0362-546X(03)00037-3
- R. Chill and M. A. Jendoubi, Convergence to steady states of solutions of non-autonomous heat equations in $\Bbb R^N$, J. Dynam. Differential Equations 19 (2007), no. 3, 777–788. MR 2350247, DOI 10.1007/s10884-006-9053-y
- F. H. Clarke, On the inverse function theorem, Pacific J. Math. 64 (1976), no. 1, 97–102. MR 425047
- Tobias Holck Colding and William P. Minicozzi II, Łojasiewicz inequalities and applications, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, Int. Press, Somerville, MA, 2015, pp. 63–82. MR 3381496, DOI 10.4310/SDG.2014.v19.n1.a3
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
- Simon Donaldson and Ed Segal, Gauge theory in higher dimensions, II, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA, 2011, pp. 1–41. MR 2893675, DOI 10.4310/SDG.2011.v16.n1.a1
- Celso Melchiades Doria, Boundary value problems for the second-order Seiberg-Witten equations, Bound. Value Probl. 1 (2005), 73–91. MR 2148375, DOI 10.1155/bvp.2005.73
- Celso Melchiades Doria, Variational principle for the Seiberg-Witten equations, Contributions to nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 66, Birkhäuser, Basel, 2006, pp. 247–261. MR 2187807, DOI 10.1007/3-7643-7401-2_{1}7
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, submitted to a refereed monograph series on September 4, 2014, arXiv:1409.1525v4, xx+475 pages.
- P. M. N. Feehan, Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions, Adv. Calc. Var. (2020), pp. 1–37. DOI https://doi.org/10.1515/acv-2020=0034. arXiv:1706.09349.
- Paul M. N. Feehan, Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections, Pacific J. Math. 200 (2001), no. 1, 71–118. MR 1863408, DOI 10.2140/pjm.2001.200.71
- Paul M. N. Feehan, Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds, Adv. Math. 312 (2017), 547–587. MR 3635819, DOI 10.1016/j.aim.2017.03.023
- Paul M. N. Feehan and Thomas G. Leness, $\rm PU(2)$ monopoles. I. Regularity, Uhlenbeck compactness, and transversality, J. Differential Geom. 49 (1998), no. 2, 265–410. MR 1664908
- Paul M. N. Feehan and Thomas G. Leness, $\rm PU(2)$ monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math. 538 (2001), 57–133. MR 1855754, DOI 10.1515/crll.2001.069
- P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces and applications to harmonic maps, submitted to a refereed journal on October 13, 2015, arXiv:1510.03817v6.
- Eduard Feireisl, Philippe Laurençot, and Hana Petzeltová, On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations 236 (2007), no. 2, 551–569. MR 2322024, DOI 10.1016/j.jde.2007.02.002
- Eduard Feireisl and Frédérique Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations 12 (2000), no. 3, 647–673. MR 1800136, DOI 10.1023/A:1026467729263
- Eduard Feireisl and Peter Takáč, Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity, Monatsh. Math. 133 (2001), no. 3, 197–221. MR 1861137, DOI 10.1007/s006050170020
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1991. MR 1081321, DOI 10.1007/978-1-4613-9703-8
- Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304, DOI 10.1007/978-3-662-03028-8
- Sergio Frigeri, Maurizio Grasselli, and Pavel Krejčí, Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differential Equations 255 (2013), no. 9, 2587–2614. MR 3090070, DOI 10.1016/j.jde.2013.07.016
- Kim A. Frøyshov, Compactness and gluing theory for monopoles, Geometry & Topology Monographs, vol. 15, Geometry & Topology Publications, Coventry, 2008. MR 2465077
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308
- Maurizio Grasselli and Hao Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal. 45 (2013), no. 3, 965–1002. MR 3048212, DOI 10.1137/120866476
- Maurizio Grasselli, Hao Wu, and Songmu Zheng, Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, SIAM J. Math. Anal. 40 (2008/09), no. 5, 2007–2033. MR 2471910, DOI 10.1137/080717833
- David Groisser and Thomas H. Parker, The geometry of the Yang-Mills moduli space for definite manifolds, J. Differential Geom. 29 (1989), no. 3, 499–544. MR 992329
- Erik Guentner, $K$-homology and the index theorem, Index theory and operator algebras (Boulder, CO, 1991) Contemp. Math., vol. 148, Amer. Math. Soc., Providence, RI, 1993, pp. 47–66. MR 1228499, DOI 10.1090/conm/148/01248
- Alain Haraux, Some applications of the Łojasiewicz gradient inequality, Commun. Pure Appl. Anal. 11 (2012), no. 6, 2417–2427. MR 2912754, DOI 10.3934/cpaa.2012.11.2417
- A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations 144 (1998), no. 2, 313–320. MR 1616968, DOI 10.1006/jdeq.1997.3393
- A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 (2007), no. 3, 449–470. MR 2328934, DOI 10.1007/s00028-007-0297-8
- Alain Haraux and Mohamed Ali Jendoubi, The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework, J. Funct. Anal. 260 (2011), no. 9, 2826–2842. MR 2772353, DOI 10.1016/j.jfa.2011.01.012
- Alain Haraux, Mohamed Ali Jendoubi, and Otared Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ. 3 (2003), no. 3, 463–484. Dedicated to Philippe Bénilan. MR 2019030, DOI 10.1007/s00028-003-1112-8
- Robert Haslhofer, Perelman’s lambda-functional and the stability of Ricci-flat metrics, Calc. Var. Partial Differential Equations 45 (2012), no. 3-4, 481–504. MR 2984143, DOI 10.1007/s00526-011-0468-x
- Robert Haslhofer and Reto Müller, Dynamical stability and instability of Ricci-flat metrics, Math. Ann. 360 (2014), no. 1-2, 547–553. MR 3263173, DOI 10.1007/s00208-014-1047-1
- A. Haydys, ${G}_2$ instantons and the Seiberg–Witten monopoles, arXiv:1607.01763.
- Andriy Haydys, The infinitesimal multiplicities and orientations of the blow-up set of the Seiberg-Witten equation with multiple spinors, Adv. Math. 343 (2019), 193–218. MR 3880858, DOI 10.1016/j.aim.2018.11.004
- Andriy Haydys and Thomas Walpuski, A compactness theorem for the Seiberg-Witten equation with multiple spinors in dimension three, Geom. Funct. Anal. 25 (2015), no. 6, 1799–1821. MR 3432158, DOI 10.1007/s00039-015-0346-3
- Joachim Hilgert and Karl-Hermann Neeb, Structure and geometry of Lie groups, Springer Monographs in Mathematics, Springer, New York, 2012. MR 3025417, DOI 10.1007/978-0-387-84794-8
- N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
- Min-Chun Hong, Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric, Ann. Global Anal. Geom. 20 (2001), no. 1, 23–46. MR 1846895, DOI 10.1023/A:1010688223177
- Min-Chun Hong and Lorenz Schabrun, Global existence for the Seiberg-Witten flow, Comm. Anal. Geom. 18 (2010), no. 3, 433–473. MR 2747435, DOI 10.4310/CAG.2010.v18.n3.a2
- Lars Hörmander, The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR 2304165, DOI 10.1007/978-3-540-49938-1
- Sen-Zhong Huang, Gradient inequalities, Mathematical Surveys and Monographs, vol. 126, American Mathematical Society, Providence, RI, 2006. With applications to asymptotic behavior and stability of gradient-like systems. MR 2226672, DOI 10.1090/surv/126
- Sen-Zhong Huang and Peter Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods, 675–698. MR 1857152, DOI 10.1016/S0362-546X(00)00145-0
- Charles Andrew Irwin, Bubbling in the harmonic map heat flow, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Stanford University. MR 2698290
- Mohamed Ali Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998), no. 1, 187–202. MR 1609269, DOI 10.1006/jfan.1997.3174
- Jürgen Jost, Xiaowei Peng, and Guofang Wang, Variational aspects of the Seiberg-Witten functional, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 205–218. MR 1386734, DOI 10.1007/BF01254344
- Jürgen Jost, Postmodern analysis, 3rd ed., Universitext, Springer-Verlag, Berlin, 2005. MR 2166001
- Dominic D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 1787733
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 407617
- Alexei Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125–160. MR 2024648, DOI 10.1515/crll.2003.097
- Klaus Kröncke, Stability of Einstein metrics under Ricci flow, Comm. Anal. Geom. 28 (2020), no. 2, 351–394. MR 4101342, DOI 10.4310/CAG.2020.v28.n2.a5
- Klaus Kröncke, Stability and instability of Ricci solitons, Calc. Var. Partial Differential Equations 53 (2015), no. 1-2, 265–287. MR 3336320, DOI 10.1007/s00526-014-0748-3
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
- Heaseung Kwon, Asymptotic convergence of harmonic map heat flow, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Stanford University. MR 2703296
- Serge Lang, Real and functional analysis, 3rd ed., Graduate Texts in Mathematics, vol. 142, Springer-Verlag, New York, 1993. MR 1216137, DOI 10.1007/978-1-4612-0897-6
- H. Blaine Lawson Jr., The theory of gauge fields in four dimensions, CBMS Regional Conference Series in Mathematics, vol. 58, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 799712, DOI 10.1090/cbms/058
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- Jiayu Li and Xi Zhang, The gradient flow of Higgs pairs, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 5, 1373–1422. MR 2825168, DOI 10.4171/JEMS/284
- Qingyue Liu and Yunyan Yang, Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds, Ark. Mat. 48 (2010), no. 1, 121–130. MR 2594589, DOI 10.1007/s11512-009-0094-4
- S. Łojasiewicz, Ensembles semi-analytiques, (1965), Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette. LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz, available at perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf.
- S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, Geometry seminars, 1982–1983 (Bologna, 1982/1983) Univ. Stud. Bologna, Bologna, 1984, pp. 115–117 (French). MR 771152
- Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542, DOI 10.1090/surv/051
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629, DOI 10.1090/coll/052
- R. B. Melrose, Lectures on pseudodifferential operators, Massachusetts Institute of Technology, November 2006, www-math.mit.edu/~rbm/Lecture_notes.html.
- John W. Morgan, Tomasz Mrowka, and Daniel Ruberman, The $L^2$-moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, II, International Press, Cambridge, MA, 1994. MR 1287851
- Tomasz Mrowka and Yann Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006), 1–69. MR 2199446, DOI 10.2140/agt.2006.6.1
- Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, Graduate Studies in Mathematics, vol. 28, American Mathematical Society, Providence, RI, 2000. MR 1787219, DOI 10.1090/gsm/028
- Christian Okonek and Andrei Teleman, The coupled Seiberg-Witten equations, vortices, and moduli spaces of stable pairs, Internat. J. Math. 6 (1995), no. 6, 893–910. MR 1354000, DOI 10.1142/S0129167X95000390
- Marco Papi, On the domain of the implicit function and applications, J. Inequal. Appl. 3 (2005), 221–234. MR 2206097, DOI 10.1155/JIA.2005.221
- Thomas H. Parker, Gauge theories on four-dimensional Riemannian manifolds, Comm. Math. Phys. 85 (1982), no. 4, 563–602. MR 677998
- V. Y. Pidstrigach and A. N. Tyurin, Localisation of Donaldson invariants along the Seiberg–Witten classes, arXiv:dg-ga/9507004.
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Johan Råde, On the Yang-Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992), 123–163. MR 1179335, DOI 10.1515/crll.1992.431.123
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- Piotr Rybka and Karl-Heinz Hoffmann, Convergence of solutions to the equation of quasi-static approximation of viscoelasticity with capillarity, J. Math. Anal. Appl. 226 (1998), no. 1, 61–81. MR 1646449, DOI 10.1006/jmaa.1998.6066
- Piotr Rybka and Karl-Heinz Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. Partial Differential Equations 24 (1999), no. 5-6, 1055–1077. MR 1680877, DOI 10.1080/03605309908821458
- D. A. Salamon, Spin geometry and Seiberg–Witten invariants, unpublished book, available at math.ethz.ch/~salamon/publications.html.
- Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703, DOI 10.2307/2006981
- Leon Simon, Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini, 1984) Lecture Notes in Math., vol. 1161, Springer, Berlin, 1985, pp. 206–277. MR 821971, DOI 10.1007/BFb0075139
- Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577, DOI 10.1090/S0894-0347-1988-0944577-9
- Ivar Stakgold and Michael Holst, Green’s functions and boundary value problems, 3rd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2011. MR 2789179, DOI 10.1002/9780470906538
- Peter Takáč, Stabilization of positive solutions for analytic gradient-like systems, Discrete Contin. Dynam. Systems 6 (2000), no. 4, 947–973. MR 1788263, DOI 10.3934/dcds.2000.6.947
- Clifford Henry Taubes, A framework for Morse theory for the Yang-Mills functional, Invent. Math. 94 (1988), no. 2, 327–402. MR 958836, DOI 10.1007/BF01394329
- P. Topping, The harmonic map heat flow from surfaces, Ph.D. thesis, University of Warwick, United Kingdom, April 1996. http://webcat.warwick.ac.uk/record=b1402741~S1.
- Peter Miles Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 (1997), no. 3, 593–610. MR 1472890
- François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, University Series in Mathematics, Plenum Press, New York-London, 1980. Pseudodifferential operators. MR 597144
- Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356
- Karen K. Uhlenbeck, Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83 (1982), no. 1, 11–29. MR 648355
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
- Vitaly Volpert, Elliptic partial differential equations. Volume 1: Fredholm theory of elliptic problems in unbounded domains, Monographs in Mathematics, vol. 101, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2778694, DOI 10.1007/978-3-0346-0537-3
- Thomas Walpuski, Gauge theory on ${G}_2$-manifolds, Ph.D. thesis, Imperial College London, United Kingdom, 2013, http://hdl.handle.net/10044/1/14365.
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
- Katrin Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2004. MR 2030823, DOI 10.4171/004
- Graeme Wilkin, Morse theory for the space of Higgs bundles, Comm. Anal. Geom. 16 (2008), no. 2, 283–332. MR 2425469
- Hao Wu and Xiang Xu, Strong solutions, global regularity, and stability of a hydrodynamic system modeling vesicle and fluid interactions, SIAM J. Math. Anal. 45 (2013), no. 1, 181–214. MR 3032974, DOI 10.1137/11085952X
- Hung Hsi Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i–xii and 289–538. MR 1079031
- Baozhong Yang, The uniqueness of tangent cones for Yang-Mills connections with isolated singularities, Adv. Math. 180 (2003), no. 2, 648–691. MR 2020554, DOI 10.1016/S0001-8708(03)00016-1
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften, vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
- Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497, DOI 10.1007/978-1-4612-0985-0