A Conformally Invariant Gap Theorem in Yang–Mills Theory

Matthew J. Gursky1, Casey Lynn Kelleher2, Jeffrey Streets3
1Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
2Department of Mathematics, Princeton University, Princeton, NJ 08540 USA
3Department of Mathematics, University of California, Irvine, CA, 92617, USA

Tóm tắt

Từ khóa


Tài liệu tham khảo

Atiyah M., Drinfeld V., Hitchin N., Manin Y.: Construction of instantons. Phys. Lett. 65, 185–187 (1978)

Aubin T.: Equations différentielles non linéaires et Probleme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)

Belavin A., Polyakov A., Schwarz A., Tyupkin Y.: Pseudoparticle solutions of the Yang–Mills equations. Phys. Lett. 59B, 8–87 (1975)

Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10, pp. xii+510. Springer, Berlin (1987)

Bor G.: Yang–Mills fields which are not self-dual. Commun. Math. Phys. 145, 393–410 (1992)

Bourguignon, J.P.: Formules de Weitzenböck en dimension 4, In: Géometrie Riemannienne de dimension 4. CEDIC, Paris (1981)

Bourguignon J.P., Lawson H.: Stability and isolation phenomena for Yang–Mills fields. Commun. Math. Phys. 79, 189–230 (1981)

Chen, Y-M., Shen, C-L.: Evolution of Yang–Mills connections, Differential geometry (Shanghair, 1991), pp. 33–41. World Sci. Publ., River Edge (1993)

Dodziuk J., Min-Oo M.: An L 2-isolation theorem for Yang–Mills fields over complete manifolds. Compos. Math. 47, 165–169 (1982)

Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Diff. Geom. 18(2), 279-315

Donaldson S.K.: Polynomial invariants for smooth four-manifolds. Topology 29, 257–315 (1990)

Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds, Oxford Mathematical Monographs, (1990)

Feehan P.: Energy gap for Yang–Mills connections, I: Four-dimensional closed Riemannian manifolds. Adv. Math. 296, 55–84 (2016)

Feehan P. Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, arXiv:1409.1525

Gerhardt C.: An energy gap theorem for Yang–Mills connections. Commun. Math. Phys. 298, 515–522 (2010)

Gursky M.J.: Four-manifolds with $${\delta W^{+} = 0}$$ and Einstein constants of the sphere. Math. Ann. 318(3), 417–431 (2000)

Gursky M.J., LeBrun C.: Yamabe invariants and spin-c structures. Geom. Funct. Anal. 8(6), 965–977 (1998)

Gursky M.J., LeBrun C.: On Einstein manifolds of positive sectional curvature. Ann. Global Anal. Geom. 17(4), 315–328 (1999)

Kozono H., Maeda Y., Naito H.: Global solution for the Yang–Mills gradient flow on 4-manifolds. Nagoya Math. J. 139, 93–128 (1998)

LeBrun C.: Ricci curvature, minimal volumes, and Seiberg–Witten theory. Invent. Math. 145(2), 279–316 (2001)

Lee J., Parker T.: The Yamabe problem. Bull. Amer. Math. Soc. 17(1), 37–91 (1987)

Min-Oo An L 2-isolation theorem for Yang–Mills fields. Comp. Math. 47, Fasc. 2, 153-163 (1982).

Parker T.: Non-minimal Yang–Mills fields and dynamics Invent. Math. 107(2), 397–420 (1992)

Parker T.: Gauge theories on four-dimensional Riemannian manifolds. Commun. Math. Phys. 85, 563–602 (1982)

Råde, J.: Decay estimates for Yang–Mills fields: two new proofs Global analysis in modern mathematics (Orono, ME, 1991) 91–105, Publish or Perish, Houston, TX (1993)

Råde J.: On the Yang–Mills heat equation in two and three dimensions. J. Reine Angew. Math. 431, 123–163 (1992)

Sadun L., Segert J.: Non-self-dual Yang–Mills connections with quadropole symmetry. Commun. Math. Phys. 145, 363–391 (1992)

Seaman, W.: Harmonic two-forms in four dimensions. Proc. Am. Math. Soc., 112(2) (1991)

Schlatter A.: Long-time behaviour of the Yang–Mills flow in four dimensions. Ann. Glob. Anal. Geom. 15, 1–25 (1997)

Shen C.L.: The gap phenomena of Yang–Mills fields over the complete manifold. Math. Z. 180, 69–77 (1982)

Sibner L.M., R.J. Sibner, K. Uhlenbeck, Solutions to Yang–Mills equations that are not self-dual, Proc. Natl. Acad. Sci. USA. 86, 8610–8613 (1989)

Simon L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. 2(118), 525–571 (1983)

Stein E.M., Weiss G.: Introduction to Fourier analysis on Euclidean spaces, Princeton Math., Series 32. Princeton University Press, Princeton, NJ (1971)

Struwe M. (1994) The Yang–Mills flow in four dimensions. Calc. Var. 2, 123–150

Xin Y.L.: Remarks on gap phenomena in four dimensions, Calc. Var. PDE. 2, 123–150 (1994)

Yang, B.: The uniqueness of tangent cones for Yang–Mills connections with isolated singularities. Adv. Math. 180(2), 648–691