Dimension Estimates for Parabolic Equations and Harmonic Maps of low Index

In the first part of this thesis we establish dimension bounds for spaces of polynomial growth solutions to divergence form parabolic partial differential equations with time-dependent coefficients as well as sharp bounds in the case of time-independent coefficients.
The second, main part of this thesis is devoted to smooth, harmonic maps of low index, in particular to the classification of such maps into round spheres.
Towards this we first study the change of the index of smooth, harmonic submersions into round spheres upon consideration as maps into higher-dimensional round spheres, that is after composition with the totally geodesic inclusion map into a higher-dimensional sphere. This is illustrated by the examples of the Hopf maps $S^3\rightarrow\,S^2$ and $S^1\rightarrow\,S^1$.
As an application we can describe for $n\geq3$ a class of harmonic morphisms of low index from the round $n$-sphere into submanifolds of the round $n$-sphere arising from smooth maps of constant rank.
Finally we prove an El Soufi-type index bound for smooth, harmonic maps from simply connected Riemannian manifolds satisfying the Killing property into round spheres. ... mehrThis bound depends only on the domain manifold, in particular it does not depend on the dimension of the codomain sphere. Focussing thereafter on the case that the codomain is the round two-sphere, we can show that every smooth, harmonic map of low index with respect to this new index bound must be a harmonic morphism.