Principal Investigator: Gang Tian
This proposal consists of three projects in integrable systems and calibrated geometry to be conducted by Emma Carberry. The first of these centers on complex tori in the nearly-Kaehler 6-sphere. Isotropic surfaces in the 6-sphere have been classified by Bryant and include all rational examples. I propose to find an algebro-geometric description of the non-isotropic complex tori. This description will involve an algebraic curve called the spectral curve, and hence give rise to a natural invariant of the torus, namely the genus of its spectral curve. The existence of tori for each value of this invariant will then be investigated and the characterization used to produce examples. Cones over these tori are associative in the imaginary octonions, which provides additional geometrical motivation for their study. The second project concerns the energy of minimal tori in the 3-sphere. Ferus, Leschke, Pedit and Pinkall have an estimate for the energy in terms of the spectral genus, which provides exciting progress toward the proof of the Willmore and Lawson conjectures, but fails to say anything about these conjectures when the spectral genus is low. It is precisely the low-genus cases upon which we shall focus, using spectral curve techniques. The final project concerns the possible singularities of special Lagrangian 3-folds, a topic of considerable current interest due to its role in mirror symmetry. Specifically, cones over tori will be studied and the question of whether the links may be real-algebraic will be addressed.
Energy is a fundamental quantity; physical systems seek to minimize energy or at least to find (possibly unstable) critical points of it. One typically studies critical points of energy, for example minimal submanifolds, which appear in all of the above proposed projects. These have a long and distinguished history in both mathematics and physics, and remain subject to much attention. Absolute energy minimizers are rare and difficult to locate. However one source of them is calibrated submanifolds, of which both associative and special Lagrangian submanifolds are examples. Special Lagrangian submanifolds have other interesting applications to physics. In super-symmetric string theory, the universe is modeled as a ten-dimensional object, consisting locally of the familiar four-dimensional space-time, and a six-dimensional space (a Calabi-Yau manifold) that is compact, and hence sufficiently 'small' that one does not observe it in everyday life. A popular approach to deepening our understanding of these Calabi-Yau manifolds is to study their special Lagrangian submanifolds.
