CHAPTER 1
Introduction
Sobolev theory on Riemannian manifolds has come into widespread usage in
modern geometry and topology. It also continues to be of great importance in
nonlinear partial differential equations (PDE's for short), variational problems, like
those in the theory of harmonic maps [26], [37] or quasiconformal deformations [32],
[35], nonlinear elasticity, continuum mechanics, and much more. Looking ahead,
we have attempted in this paper to present such mappings with all their nuances
and possible applications.
The primary objects of our study are weakly differentiable mappings:
(1.1) / : X ^ Y
where X and Y are smooth compact oriented Riemannian manifolds without bound-
ary, dimX = n ^ 2 and dimY = m 2. One might say that C. B. Morrey [43]
was the first to consider such mappings. The Sobolev class
W1,P(K,
Y) can be
defined in a myriad of ways that are not always equivalent. In our approach we
appeal to the celebrated theorem of J. Nash [47], which ensures that Y can be
c€°°-
isometrically imbedded in some Euclidean space R^. Let us assume that Y C R^,
for simplicity. This being so, we say that / = (f1,..., fN) : X RN belongs to the
Sobolev space W1,P(X, Y) if each coordinate function fl : X R lies in the usual
Sobolev space #'1'P(X), and f(x) G Y for almost every x G l We do not reserve
any particular notation of the Riemannian tensors on X and Y, as these tensors will
be fixed for the duration of this paper. The volume elements on X and Y, denoted
by dx G
^°°(AnX)
and dy G
^°°(AmY),
will be the ones induced by the orientation
and the metric tensors. In this way
W1,P(H,
Y), 1 ^ p oo, becomes a complete
metric subspace of the linear space
W1,PQL^ WN).
In the Riemannian manifolds setting it is not clear at all whether smooth map-
pings / G ^°°(X, Y) are dense in W1,P(X, Y), a question raised by J. Eells and
L. Lemaire [10]. This is trivially the case for p n. R. Schoen and K. Uhlenbeck
[49], [50] showed that the answer is also positive when p n. That is all we can
have in the category of the Sobolev spaces
W1,P(X,
Y), unless additional topolog-
ical conditions are imposed on the manifolds X and Y [23], [24]. For example, in
the same paper R. Schoen and K. Uhlenbeck [50] demonstrate that ^ ( S
7 1
, §
n_1
)
is not dense in
W1,p(Bn,
S
n_1
) for every n—1 ^ p n. While it is not clear at this
point, the Sobolev space #^1,n(X, Y), with n dimX ^ 2, will be the borderline
case for many more phenomena concerning weakly different iable mappings. Other
related papers are [2], [3] [17], [18], [19], [20]. Sobolev spaces with exponents
1 p n are natural in the theory of harmonic mappings [26], [10], [37], [49]
and other related areas. However, properties of these mappings are very different
from those in
#^1,n(X,
Y). This difference lies fairly deep in the concept of the
topological degree. If dimX = dimY = n, then a smooth mapping / : X Y has
l
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