In the present work, a coordinate-free way is
suggested to handle conformal maps of a Riemannian surface into a space of
constant curvature of maximum dimension 4, modeled on the non-commutative field
of quaternions. This setup for the target space and the idea to treat differential
2-forms on Riemannian surfaces as quadratic functions on the tangent space,
are the starting points for the development of the theory of conformal maps and
in particular of conformal immersions. As a first result, very nice conditions
for the conformality of immersions into 3- and 4-dimensional space-forms are
deduced and a simple way to write the second fundamental form is found.
If the target space is euclidean 3-space, an
alternative approach is proposed by fixing a spin structure on the Riemannian
surface. The problem of finding a local immersion is then reduced to that of
solving a linear Dirac equation with a potential whose square is the Willmore
integrand. This allows to make statements about the structure of the moduli
space of conformal immersions and to derive a very nice criterion for a
conformal immersion to be constrained Willmore. As an application the Dirac
equation with constant potential over spheres and tori is solved. This yields explicit
immersion formulae out of which there were produced pictures, the Dirac-spheres
and -tori. These immersions have the property that their Willmore integrand
generates a metric of vanishing and constant curvature, respectively.
As a next step an affine immersion theory is
developped. This means, one starts with a given conformal immersion into
euclidean 3-space and looks for new ones in the same conformal class. This is
called a spin-transformation and it leads one to solve an affine Dirac
equation. Also, it is shown how the coordinate-dependent generalized
Weierstrass representation fits into the present framework. In particular, it
is now natural to consider the class of conformal immersions that admit new
conformal immersions having the same potential. It turns out, that all geometrically
interesting immersions admit such an isopotential spin-transformation and that
this property of an immersion is even a conformal invariant of the ambient
space.
It is shown that conformal isothermal immersions generate
both via their dual and via Darboux transformations non-trivial families of
new isopotential conformal immersions. Similarly to this, conformal
(constrained) Willmore immersions produce non-trivial families of isopotential
immersions of which subfamilies are (constrained) Willmore again having even
the same Willmore integral. Another observation is, that the Euler-Lagrange
equation for the Willmore problem is the integrability condition for a
quaternionic 1-form, which generates a conformal minimal immersions into
hyperbolic 4-space. Vice versa, any such immersion determines a conformal
Willmore immersion. As a consequence, there is a one-to-one correspondence
between conformal minimal immersions into Lorentzian space and those into
hyperbolic space, which generalizes to any dimension. There is also induced an
action on conformal minimal immersions into hyperbolic 4-space. Another fact
is, that conformal constant mean curvature (cmc) immersions into some
3-dimensional space form unveil to be isothermal and constrained Willmore. The
reverse statement is true at least for tori. Finally a very simple proof of a
theorem by R.Bryant concerning Willmore spheres is given.
In the last part, time-dependent conformal
immersions are considered. Their deformation formulae are computed and it is
investigated under what conditions the flow commutes with Moebius transformations.
The modified Novikov-Veselov flow is written down in a conformal invariant way
and explicit deformation formulae for the immersion function itself and all of
its invariants are given. This flow commutes with Moebius transformations. Its
definition is coupled with a delta-bar problem, for which a solution is
presented under special conditions. These are fulfilled at least by cmc
immersions and by surfaces of revolution and the general flow formulae reduce
to very nice formulae in these cases.