theory of differential forms on manifolds

Author:
Chad Scott

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2075-2096

MSC:
Primary 58A14; Secondary 58G03

MathSciNet review:
1297538

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we establish a Hodge-type decomposition for the space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an estimate which contains, as a special case, the result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the boundedness of Green's operator which we use in developing the theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the -harmonic equation.

**[C]**Henri Cartan,*Differential forms*, Translated from the French, Houghton Mifflin Co., Boston, Mass, 1970. MR**0267477****[Co]**J.B. Conway,*A course in functional analysis*(2nd ed.), Macmillan, New York, 1988.**[Con]**P. E. Conner,*The Neumann’s problem for differential forms on Riemannian manifolds*, Mem. Amer. Math. Soc.**No. 20**(1956), 56. MR**0078467****[D]**G. F. D. Duff,*Differential forms in manifolds with boundary*, Ann. of Math. (2)**56**(1952), 115–127. MR**0048136****[Da]**Bernard Dacorogna,*Direct methods in the calculus of variations*, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR**990890****[DS]**S. K. Donaldson and D. P. Sullivan,*Quasiconformal 4-manifolds*, Acta Math.**163**(1989), no. 3-4, 181–252. MR**1032074**, 10.1007/BF02392736**[DSp]**G. F. D. Duff and D. C. Spencer,*Harmonic tensors on Riemannian manifolds with boundary*, Ann. of Math. (2)**56**(1952), 128–156. MR**0048137****[EG]**Lawrence C. Evans and Ronald F. Gariepy,*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660****[F]**Harley Flanders,*Differential forms with applications to the physical sciences*, 2nd ed., Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1989. MR**1034244****[GT]**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****[H]**Piotr Hajlasz,*Note on Meyers-Serrin's Theorem*, Expositiones Math. (to appear).**[I]**Tadeusz Iwaniec,*𝑝-harmonic tensors and quasiregular mappings*, Ann. of Math. (2)**136**(1992), no. 3, 589–624. MR**1189867**, 10.2307/2946602**[IL]**Tadeusz Iwaniec and Adam Lutoborski,*Integral estimates for null Lagrangians*, Arch. Rational Mech. Anal.**125**(1993), no. 1, 25–79. MR**1241286**, 10.1007/BF00411477**[IM]**Tadeusz Iwaniec and Gaven Martin,*Quasiregular mappings in even dimensions*, Acta Math.**170**(1993), no. 1, 29–81. MR**1208562**, 10.1007/BF02392454**[K]**Kunihiko Kodaira,*Harmonic fields in Riemannian manifolds (generalized potential theory)*, Ann. of Math. (2)**50**(1949), 587–665. MR**0031148****[LM]**H. Blaine Lawson Jr. and Marie-Louise Michelsohn,*Spin geometry*, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR**1031992****[M]**Charles B. Morrey Jr.,*Multiple integrals in the calculus of variations*, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR**0202511****[Mo]**Marston Morse,*Global variational analysis*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1976. Weierstrass integrals on a Riemannian manifold; Mathematical Notes, No. 16. MR**0494242****[MM]**Sunil Mukhi and N. Mukunda,*Introduction to topology, differential geometry and group theory for physicists*, Wiley Eastern Limited, New Delhi, 1990.**[N]**Raghavan Narasimhan,*Analysis on real and complex manifolds*, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR**0251745****[RRT]**Joel W. Robbin, Robert C. Rogers, and Blake Temple,*On weak continuity and the Hodge decomposition*, Trans. Amer. Math. Soc.**303**(1987), no. 2, 609–618. MR**902788**, 10.1090/S0002-9947-1987-0902788-8**[R]**H. L. Royden,*Real analysis*, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR**0151555****[S]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[Sc]**C.H. Scott,*theory of differential forms on manifolds*, Ph.D. Thesis, Mathematics Department, Syracuse University, June, 1993.**[SS]**L. M. Sibner and R. J. Sibner,*A non-linear Hodge-de-Rham theorem*, Acta Math.**125**(1970), 57–73. MR**0281231****[U]**K. Uhlenbeck,*Regularity for a class of non-linear elliptic systems*, Acta Math.**138**(1977), no. 3-4, 219–240. MR**0474389****[W]**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****[Z]**Eberhard Zeidler,*Nonlinear functional analysis and its applications*, Springer-Verlag, New York, 1990.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58A14,
58G03

Retrieve articles in all journals with MSC: 58A14, 58G03

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1297538-7

Keywords:
Differential form,
Hodge decomposition,
harmonic integral,
Sobolev space

Article copyright:
© Copyright 1995
American Mathematical Society