RUI: Flows of G2-Structures on Manifolds

The University Of Texas Rio Grande Valley, Edinburg TX

Investigators

Abstract

One of the most enduring scientific problems is to understand the properties of the universe and the laws that govern it. Ever since Einstein's theory of general relativity, it was understood that there is a very close link between the physics and the geometry of the universe. However, most recent physical theories, such as superstring theory and M-theory, show that the physical properties of 4-dimensional spacetime may be described in terms of the geometry of hidden six- or seven-dimensional spaces. In this project, the PI will study the properties of a particular kind of seven-dimensional spaces which appear in these theories, known as manifolds with a G2-structure. The main goal of this project is to further develop the theory of deformations or `flows' of G2-structures which will effect smooth transitions between different classes of G2-structures. The PI anticipates that this will pave the way towards giving sufficient conditions for existence of torsion-free G2-structures, which is one of the most significant open problems in differential geometry. (A G2-structure defines, at each point of the manifold, a `dot product' of vectors -- a way of multiplying two vectors to get a scalar -- as well as a `cross product' that allows one to multiply two vectors to produce a third vector. Torsion-free G2-structures are those for which these dot and cross products are maximally compatible.) As part of the project, the PI will train undergraduate research assistants and will involve them in the activities of the Experimental Algebra & Geometry Lab, of which the PI is a co-director. The lab's activities will combine training, mentoring, research, and outreach to promote mathematics at various levels -- starting at K-12, the broader community, and going all the way to graduate level. In his prior work, the PI introduced the modified Laplacian coflow of co-closed G2-structures that rectified the non-parabolicity of the standard Laplacian coflow. The PI also introduced a description of G2-structures in terms of octonion bundles that interpreted the torsion of a G2-structure as an octonionic connection and the choice of a G2-structure within the same metric class as a choice of gauge. G2-structures with divergence-free torsion were then interpreted as critical points of an energy functional and as an analogue of the Coulomb gauge. This project builds upon this prior work. The main goal is to prove existence properties for the standard Laplacian coflow. This will be accomplished by first proving the existence of G2-structures with divergence-free torsion within a fixed metric class using harmonic map techniques. This will then be used as a gauge-fixing condition for the Laplacian coflow, which will relate it to the modified coflow for which existence is known. The second objective of this project is to study and construct other significant flows of G2-structures, with the aim of proving existence and stability under appropriate conditions and analyzing the behavior of these flows on homogeneous manifolds. The third objective of this project is to study the properties of octonion-bundle-valued differential forms and to obtain an octonion-valued analogue of Dolbeault cohomology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.