» Jost J. - The geometric calculus of variations: a short survey and a list of open problems
Jost J. - The geometric calculus of variations: a short survey and a list of open problems
||The geometric calculus of variations: a short survey and a list of open problems
||Since my presentation of open problems at the Oberwolfach meeting on the "Calculus of variations" in April 1986 met with much interest from the side of the participants, it seemed useful to have the present notes available in written form.
The aim is to describe some of the major open problems in the area and thereby to show the potential and the limits of the mathematical methods presently available. Although some of the problems seem too hard to be solvable in the immediate future, I believe that even for a beginner it is very useful to know what the essential difficulties in a specific area of mathematics are. I also tried to describe some background for some of the problems.
Many of the problems are wellknown, and sometimes I was unable to find a specific origin or reference for a problem. Many of the problems also already appear in other problem lists. Because of the scope of the present list and since of course I could update and expand some of the older presentations, I nevertheless believe that my collection represents a useful documentation of the present state of our insight into the geometric aspects of the calculus of variations.
Before discussing the individual problem, I shall give a general overview of the geometric calculus of variations, in order to provide the framework within which the problems acquire their meaning. In this overview, I shall focus on area, energy, and curvature integrals, although the problem list has a somewhat broader scope, in order to point out the major lines of developments, the guiding underlying principles, and the interplay of geometry and analysis that is characteristic for the subject.
My main sources for the problems were Yau's differential geometric problem section in [Y2] (an updated version is under preparation), the list of problems on harmonic