theory of differential forms on manifolds

Author:
Chad Scott

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

MSC:
Primary 58A14; Secondary 58G03

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

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]**H. Cartan,*Differential forms*, Houghton Mifflin, Boston, MA, 1970. MR**0267477 (42:2379)****[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.**20**(1956). MR**0078467 (17:1197e)****[D]**G.F.D. Duff,*Differential forms on manifolds with boundary*, Ann. of Math.**56**(1952), 115-127. MR**0048136 (13:986e)****[Da]**B. Dacorongna,*Direct methods in the calculus of variations*, Springer-Verlag, New York, 1989. MR**990890 (90e:49001)****[DS]**S.K. Donaldson and D.P. Sullivan,*Quasiconformal**-manifolds*, Acta Math.**163**(1989), 181-252. MR**1032074 (91d:57012)****[DSp]**G.F.D. Duff and D.C. Spencer,*Harmonic tensors on Riemannian manifolds with boundary*, Ann. of Math.**56**(1952), 128-156. MR**0048137 (13:987a)****[EG]**Lawrence C. Evans and Ronald F. Gariepy,*Measure theory and fine properties of functions*, CRC Press, Boca Raton, FL, 1992. MR**1158660 (93f:28001)****[F]**Harley Flanders,*Differential forms with applications to the physical sciences*, Dover, New York, 1989. MR**1034244 (90k:53001)****[GT]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*(2nd ed.), Springer-Verlag, New York, 1983. MR**737190 (86c:35035)****[H]**Piotr Hajlasz,*Note on Meyers-Serrin's Theorem*, Expositiones Math. (to appear).**[I]**T. Iwaniec,*-harmonic tensors and quasiregular mappings*, Ann. of Math.**136**(1992), 589-624. MR**1189867 (94d:30034)****[IL]**T. Iwaniec and A. Lutoborski,*Integral estimates for null Lagrangians*, Arch. Rational Mech. Anal. (to appear). MR**1241286 (95c:58054)****[IM]**T. Iwaniec and G. Martin,*Quasiregular mappings in even dimensions*, Acta Math.**170**(1993), 29-81. MR**1208562 (94m:30046)****[K]**K. Kodaira,*Harmonic fields in Riemannian manifolds*, Ann. of Math.**50**(1949), 587-665. MR**0031148 (11:108e)****[LM]**H.B. Lawson and M.L. Michelsohn,*Spin geometry*, Princeton Univ. Press, Princeton, NJ, 1989. MR**1031992 (91g:53001)****[M]**C.B. Morrey,*Multiple integrals in the calculus of variations*, Springer-Verlag, Berlin, 1966. MR**0202511 (34:2380)****[Mo]**Marston Morse,*Global variational analysis, Weierstrass integrals on a Riemannian manifold*, Princeton Univ. Press, Princeton, NJ, 1976. MR**0494242 (58:13150)****[MM]**Sunil Mukhi and N. Mukunda,*Introduction to topology, differential geometry and group theory for physicists*, Wiley Eastern Limited, New Delhi, 1990.**[N]**R. Narasimhan,*Analysis on real and complex manifolds*, Elsevier Science B.V., Amsterdam, 1968. MR**0251745 (40:4972)****[RRT]**J.W. Robbin, R.C. Rogers and B. Temple,*On weak continuity and the Hodge decomposition*, Trans. Amer. Math. Soc.**303**(1987), 609-618. MR**902788 (88m:58004)****[R]**H.L. Royden,*Real analysis*, (3rd ed.), Macmillan, New York, 1988. MR**0151555 (27:1540)****[S]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Univ. Press, Princeton, NJ, 1970. MR**0290095 (44:7280)****[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.B. Sibner,*A non-linear Hodge De Rham Theorem*, Acta Math.**125**(1970), 57-73. MR**0281231 (43:6950)****[U]**K. Uhlenbeck,*Regularity for a class of nonlinear elliptic systems*, Acta Math.**138**(1977), 219-250. MR**0474389 (57:14031)****[W]**Frank W. Warner,*Foundations of differentiable manifolds and Lie groups*, Springer-Verlag, New York, 1983. MR**722297 (84k:58001)****[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