Isoperimetric Inequalities and Rigidity

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There are two major closely related themes to this project. The first is the study of rigidity theorems (i.e. metric uniqueness) on compact manifolds with and without boundary. The second is the study of sharp isoperimetric inequalities. Here "isoperimetric inequality" should be interpreted as an inequality between global geometric quantities (such as volume, volume of the boundary, eigenvalues of the Laplace Beltrami operator, lengths of closed geodesics etc.) while sharp means that the inequalities are the best possible. They are related in that the case of equality in appropriate sharp isoperimetric inequalities can lead to interesting rigidity results.We consider for example isospectral problems: to what extent must spaces with the same spectra (e.g. eigenvalues of the Laplace Beltrami operator, or Lengths of closed geodesics) be isometric. This also includes questions about metric rigidity induced by conjugacy of geodesic flows, as well as inverse scattering problems. Another theme of the proposal is the study of infinite groups G acting cocompactly on nonpositively curved spaces H (in the sense of Alexandrov). The project is to study the relationship between the geometry of H and the induced action of G on the ideal boundary of H. This is an aspect of geometricgroup theory. The rigidity theme of the project concerns the question of whether a space can be determined by a certain set of data. One part of this relates to questions of remote sensing. For example: can you determine the density of an object (say a person's body or the moon) from measurements taken "from the outside"? The CAT scan is a practical example where one determines the mass density (or more accurately the absorption coefficient) of an object from measurements of the total mass along straight lines. An alternative set of measurements is the set of times it takes for sound to travel between any two points on the boundary (this is a special case of the boundary rigidity question dealt with in the proposal). A related set of measurements is to record the exit times and directions of geodesics given their entry directions (this is the "geodesic lens" or "scattering" data). The thrust of the proposed study is to determine under which circumstances certain sets of data (e.g. eigenvalues, lengths of closed geodesics, distances between boundary points, lens data) are sufficient to completely determine the geometry of the spaces in question. In some cases it is non-uniqueness that is interesting. For example, in cloaking the goal is to make it the space in question (the object to be cloaked) appear from the outside like a different space (empty space).