The proposed project is concerned with the use of analytical tools to understand the qualitative behavior of nonlinear partial differential equations arising in both differential geometry and mathematical physics. For example, we intend to study harmonic maps from higher dimensional Riemannian manifolds into general target manifolds. We also plan to study nonlinear wave equations, using methods from harmonic analysis as well as vector field methods originating in differential geometry. In addition, we aim for a general convergence result for the Yamabe flow in conformal geometry. All known results in this direction either assume the manifold to be locally conformally flat (such as the work of R. Ye), or they require a rather restrictive bound on the initial energy (such as the recent work of M. Struwe and H. Schwetlick). It is known that shortly before a singularity the solution must look like a superposition of "peak solutions", whose asymptotic profile is explicitly known. A careful analysis of the interaction between different peaks suggests that it costs energy to make the peak higher, i.e. to concentrate the energy on a smaller region. Since the evolution equation is designed to decrease the energy, this indicates that no singularities should form. While the result is known for initial energy less than two peaks, the general case offers a more interesting picture.
The questions to be studied in this project are mainly motivated by differential geometry. However, it may seem surprising that many of these equations also play an important role in applied sciences. For example, harmonic maps into the two-dimensional sphere are closely related to the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum. Moreover, the Yamabe flow in conformal geometry can be reduced to the fast diffusion case of the porous medium equation. In the case of positive scalar curvature, the effect of the reaction term is opposed to that of the diffusion term, and it is a non-trivial issue to decide which of these effects will prevail.
