Geometric Variational Problems from String Theory

In string theory one studies the dynamics of one-dimensional objects, called strings, which propagate through a curved space. In analogy to point particles, which propagate on the shortest distance between two points, one requires that the area swept out by the string is minimal. Such kind of problems have been investigated by mathematicians, in particular geometers, for a long time: A minimal surface is a surface in a (curved) space, which has minimal area. A classical example for minimal surfaces are soap films, which are clamped on a frame. In order to deal with this problem mathematically, one applies the calculus of variation: One considers the so-called area functional, which associates a number (area) to every surface in space. Employing the methods of geometric analysis one can find those surfaces, which minimize the surface area. These are called critical points of the area functional. One big advantage of the area functional is the fact, that it is bounded from below since the area of a surface cannot be negative. For such kind of variational problems there exist powerful mathematical tools. The variational problems that arise in string theory are also formulated in terms of differential geometry. However, these are more complicated since they are unbounded from below. The first part of the project focuses on the investigation of geometric and analytic properties of various variational problems from string theory. To handle the unbounded functionals it will be necessary to develop new mathematical regularization methods. These transform unbounded functionals to bounded ones, for which a large number of mathematical tools exist. The difficult question will be whether the "regularized problem" also provides information on the original problem. In the second part of the project we want to study the equations that govern the dynamics of a superstring in a curved space. Formally, these equations comprise of a linear and a non-linear wave equation. Linear wave equation model the unperturbed expansion of waves, i.e. sound waves. On the other hand, the solutions of non-linear wave equations may develop singularities: In the case of sound waves these would correspond to the occurrence of supersonic waves. For both equations that govern the dynamics of a superstring there already exists an extensive number of results in the mathematical literature both in analysis and geometry. In the course of this project we want to cleverly combine and extend the existing methods. The nonlinearities appearing here are manageable from a mathematical point of view. For this reason one can expect that the equations can have both global solutions, but also solutions that develop a singularity. It will be exciting to investigate in which cases singularities will occur. In addition, we want to explore how the geometry of the surrounding space influences the dynamics of the superstring.

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