Abstract
Let F : [0, ∞) → [0, ∞) be a strictly increasing C 2 function with F(0) = 0. We unify the concepts of F-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce F-Yang-Mills fields, F-degree, F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When \({F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,}\) and \({1-\sqrt{1-2t},}\) the F-Yang-Mills field becomes an ordinary Yang-Mills field, p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the E F,g −energy functional (resp. F-Yang-Mills functional) and derive the first variational formula of the E F,g −energy functional (resp. F-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors are naturally linked to F-conservation laws and yield monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry. Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory, a “macroscopic” version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for p−forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for F−harmonic maps (which include harmonic maps, p-harmonic maps, exponentially harmonic maps, minimal graphs and maximal space-like hypersurfaces, etc.), F−Yang-Mills fields, extended Born-Infeld fields, and generalized Yang-Mills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds, etc. As another consequence, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains for vector bundle-valued 1-forms satisfying an F-conservation law, generalizing and refining the work of Karcher and Wood on harmonic maps. We also obtain generalized Chern type results for constant mean curvature type equations for p−forms on \({\mathbb{R}^m}\) and on manifolds M with the global doubling property by a different approach. The case p = 0 and \({M=\mathbb{R}^m}\) is due to Chern.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Allard W.K.: On the first variation of a varifold. Ann. of Math. 95(2), 417–491 (1972)
Almgren F.J. Jr: Some interior regularity theorems for minimal surfaces and extension of Bernstein’s theorem. Ann. of Math. 84(2), 277–292 (1966)
Ara M.: Geometry of F−harmonic maps. Kodai Math. J. 22, 243–263 (1999)
Baird P.: Stress-energy tensors and the Lichnerowicz Laplacian. J. Geom. Phys. 58(10), 1329–1342 (2008)
Bernstein, S.: Sur un theoreme de geometrie at ses application aux equations aux derivees partielles du type elliptique. Comm. Soc. Math. Kharkov 15(2), 38–45 (1915-C1917)
Baird, P., Eells, J.: A conservation law for harmonic maps. In: Geometry Symposium, Utrecht 1980, in: Lecture notes in Mathematics, Vol. 894, Berlin, Heidleberg-NewYork: Springer, 1982, pp. 1–25
Born M., Infeld L.: Foundation of a new field theory. Proc. R. Soc. London Ser. A. 144, 425–451 (1934)
Barbashov, B.M., Nesterenko, V.V.: Introduction to the relativistic string theory. Singapore World Scientific, 1990
E. Calabi, Examples of Bernstein problems for some nonlinear equations. In: Global Analysis, (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Providence, RI: Amer. Math. Soc., 1970, pp. 223–230
Chen Q.: Stability and constant boundary-value problems of harmonic maps with potential. J. Aust. Math. Soc. (Series A) 68, 145–154 (2000)
Chern S.S.: On the curvature of a piece of hypersurface in Euclidean space. Abh. Math. Sem. Hamburg 29, 77–91 (1965)
Chen Q., Zhou Z.R.: On gap properties and instabilities of p-Yang-Mills fields. Canad. J. Math. 59(6), 1245–1259 (2007)
Deser S., Gibbons G.W.: Born-Infeld actions. Class. Quantun 15, L35-9 (1998)
Ding, Q.: The Dirichlet problem at infinity for manifolds of nonpositive curvature. In: Gu, C. H. (ed.) et al., Differential geometry, Proc. of the Symp. in honour of Prof. Su Buchin on his 90th birthday (Shanghai), Singapore: World Sci., 1993, pp. 48–58
Dong Y.X.: Bernstein theorems for spacelike graphs with parallel mean curvature and controlled growth. J. Geom. Phys. 58(3), 324–333 (2008)
Escobar J.F., Freire A.: The spectrum of the Laplacian of manifolds of positive curvature. Duke Math. J. 65, 1–21 (1992)
Escobar J.F., Freire A.: The differential form spectrum of manifolds of positive curvature. Duke Math. J. 69, 1–41 (1993)
Escobar J.F., Freire A., Min-Oo M.: L 2 vanishing theorems in positive curvature. Indiana Univ. Math. J. 42(4), 1545–1554 (1993)
Federer H., Fleming W.H.: Normal and integral currents. Ann. of Math. 72, 458–520 (1960)
Fleming W.H.: On the oriented Plateau problem, Rend. Circ. Mat. Palermo 11(2), 69–90 (1962)
de Giorgi E.: Una estensione del theorema di Bernstein. Ann. Scuola Norm. Sup. Pisa 19(3), 79–85 (1965)
Garber W.D., Ruijsenaars S.N.M., Seiler E., Burns D.: On finite action solutions of the nonlinear σ−model. Ann. Phys. 119, 305–325 (1979)
Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. Lecture Notes in Math. 699, Berlin, Heidleberg-New York:Springer-Verlag, 1979
Hardt R, Lin F.H.: Mappings minimizing the L p norm of the gradient. XL Comm. Pure App. Math. 40(5), 555–588 (1987)
Hildebrandt S.: Liouville theorems for harmonic mappings, and an approach to Bernstein theorems. Ann. Math. Stud. 102, 107–131 (1982)
Hu H.S.: On the static solutions of massive Yang-Mills equations. Chinese Annals of Math. 3, 519–526 (1982)
Hu H.S.: A nonexistence theorem for harmonic maps with slowly divergent energy. Chinese Ann. of Math. Ser. B 5(4), 737–740 (1984)
Jaffe A, Taubes C.: Vortices and Monopoles: Structures of Static Gauge Theories. Birkhauser, Boston (1980)
Kassi M.: A Liouville Theorem for F−harmonic maps with finite F−energy. Electronic J. Diff. Eqs. 15, 1–9 (2006)
Ketov, S.V.: Many faces of Born-Infeld theory. http://arxiv.org/abs/hep-th/0108189v1, 2001
Karcher H., Wood J.C.: Non-existence results and growth properties for harmonic maps and forms. J. Reine Angew. Math. 353, 165–180 (1984)
Lawson, H.B. Jr.: The theory of gauge fields in four dimensions. CBMS Regional Conference Series in Mathematics 58, Providence, RI: Amer. Math. Soc., 1985
Lin F.H., Yang Y.S.: Gauged harmonic maps, Born-Infeld electromagnetism, and Magnetic Vortices. Comm. Pure. Appl. Math. LVI, 1631–1665 (2003)
Liu J.C.: Vanishing theorem for L p−forms valued on vector bundle (Chinese). J. of Northwest Normal University (Natural Science) 38(4), 20–22 (2004)
Liu J.C.: Constant boundary-valued problems for p− harmonic maps with potential. J. Geom. Phys. 57, 2411–2418 (2007)
Liu, J.C., Liao, C.S.: The energy growth property for p− harmonic maps. J. East China Normal Univ. (Natural Sci.) no. 2, 19–23 (2004)
Liu J.C., Liao C.S.: Nonexistence Theorems for F− harmonic maps (Chinese). J. Math. Research and Exposition 26, 311–316 (2006)
Lu, M., Shen, X.W., Cai, K.R.: Liouville type Theorem for p−forms valued on vector bundle (Chinese). J. Hangzhou Normal Univ. (Natural Sci.) 7, 96–100 (2008)
Luckhaus S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J. 37, 349–367 (1988)
Li P., Wang J.P.: Finiteness of disjoint minimal graphs. Math. Res. Lett. 8, 771–777 (2001)
Li J.F., Wei S.W.: A p-harmonic approach to generalized Bernstein problem. Commun. Math. Anal. Conference 1, 35–39 (2008)
Lee Y.I., Wang A.N., Wei S.W.: On a generalized 1-harmonic equations and the inverse mean curvature flow. J. Geom. Phys. 61(2), 453–461 (2011)
Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm Pure App. Math. XIV, 577–591 (1961)
Rigoli M., Setti A.: Energy estimates and Liouville theorems for harmonic maps. Intern. J. Math. 11(3), 413–448 (2000)
Price, P., Simon, L.: Monotonicity formulae for Harmonic maps and Yang-Mills fields. preprint, Canberra 1982. Final verson by P. Price, Amonotonicity formula for Yang-Mills fields, Manus. Math. 43, 131–166 (1983)
Salavessa I.M.: Graphs with parallel mean curvature. Proc. A.M.S. 107(2), 449–458 (1989)
Salavessa I.M.: Spacelike graphs with parallel mean curvature. Bull. Bel. Math. Soc. 15, 65–76 (2008)
Sanini A.: Applicazioni tra variet‘a riemanniane con energia critica rispetto a deformazioni di metriche. Rend. Mat. 3, 53–63 (1983)
Sealey H.C.J.: Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory. Math. Soc. Camb. Phil. Soc. 91, 441–452 (1982)
Sealey, H.C.J.: The stress energy tensor and vanishing of L 2 harmonic forms. Preprint, 1983
Sibner L., Sibner R., Yang Y.S.: Generalized Bernstein property and gravitational strings in Born-Infeld theorey. Nonlinearity 20, 1193–1213 (2007)
Simons J.: Minimal varieties in Riemannian manifolds. Ann. of Math. 88(2), 62–105 (1968)
Schoen R., Uhlenbeck K.: A regularity theory for harmonic maps. J. Diff. Geom. 17, 307–335 (1982)
Wei, S.W.: On 1-harmonic functions, SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 127, (2007) 10 pp.
Wei S.W.: p-Harmonic geometry and related topics. Bull. Transilv. Univ. Brasov Ser. III 1(50), 415–453 (2008)
Wei S.W.: Representing homotopy groups and spaces of maps by p-harmonic maps. Indiana Univ. Math. J. 47, 625–670 (1998)
Xin Y.L.: Differential forms, conservation law and monotonicity formula. Scientia Sinica (Ser A) XXIX, 40–50 (1986)
Xin Y.L.: On Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comm. Math. Helv. 66, 590–598 (1991)
Xin Y.L.: A rigidity theorem for a space-like graph of higher codimension. Manus. Math. 103(2), 191–202 (2000)
Yang Y.S.: Classical solutions in the Born-Infeld theory. Proc. R. Soc. Lond. A 456, 615–640 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N.A. Nekrasov
Supported by NSFC grant No 10971029, and NSFC-NSF grant No 1081112053.
Research was partially supported by NSF Award No DMS-0508661, the OU Presidential International Travel Fellowship, and the OU Faculty Enrichment Grant.
Rights and permissions
About this article
Cite this article
Dong, Y., Wei, S.W. On Vanishing Theorems for Vector Bundle Valued p-Forms and their Applications. Commun. Math. Phys. 304, 329–368 (2011). https://doi.org/10.1007/s00220-011-1227-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1227-8