Generalizations of (hyper-)Kähler geometry and geometric flows related to Ricci-flat Riemannian manifolds
Final Report Abstract
In the first part of the project, classification results for left-invariant SKT structures on two-step solvable Lie algebras were obtained in cooperation with Andrew Swann using the shear construction which was introduced in joint work with him before the start of the project. More exactly, the SKT structures among the almost-Hermitian structures on two-step solvable Lie alebras with commutator ideal of low dimension or codimension or with commutator ideal being totally real (and some extra conditions) were identified. Moreover, all six-dimensional two-step solvable SKT Lie algebras were classified up to one particular class. Besides, in the first part of the project, also a classification of all complex symplectic Lie algebras in dimension 8 and some interesting examples of compact complex symplectic nilmanifolds were obtained in joint work with G. Bazzoni, A. Latorre and B. Meinke. This classification result was achieved by applying a new construction invented by ourselves, the so-called complex symplectic oxidation. This construction builds from a complex symplectic Lie algebra and certain “oxidation data” a complex symplectic Lie algebra of four more real dimensions. It is to be expected that this construction will have further applications. Such applications as well as other open questions, e.g., a geometric interpretation of complex symplectic oxidation, are currently discussed in a follow-up project. The second part of the project was planned to be on different open problems about geometric flows related to Riemannian manifolds of exceptional holonomy but eventually the goals changed during the course of the funding period: As planned, the spinor flows on homogeneous manifold was studied together with H. Weiß and L. Schiemanowski. Here, the spinor flows were extended to non-compact homogeneous manifolds, many explicit homogeneous solutions were obtained and also some interesting results on the long-time behaviour of these flows on (certain types of) low-dimensional homogeneous manifolds and on the homogeneous stability of certain critical points. In contrast, the other geometric flows, which were mentioned in the project description, were not investigated further. Instead, the first example of an Einstein non-Ricci-flat pseudo-Riemannian metric (of signature (3,4)) on a seven-dimensional nilpotent Lie algebra (by results of Conti and Rossi the lowest possible dimension) was constructed in cooperation with M. Fernández and J. Sánchez. It was shown that this particular pseudo-metric, as well as any other Einstein non-Ricci-flat pseudo-metric on that particular Lie algebra, cannot be induced by a closed G2*-structure. Moreover, further results on closed G2*-structures were obtained. Currently, other open question (e.g., a classification of all such pseudo-metrics in dimension 7) are investigated. Moreover, in cooperation with Simon Salamon, it was shown that closed left-invariant G2-eigenforms for the Laplacian operator (with non-zero eigenvalue) cannot exist on seven-dimensional almost nilpotent Lie groups G and that exact G2-structures cannot exist on compact solvmanifolds arising from such Lie groups G. The proof is based on a reduction to the six-dimensional nilpotent ideal and yields other interesting results on certain six-dimensional structures. The seven-dimensional results are (one of) the first known results on the open questions if there exists closed G2-eigenforms at all and if a compact manifold may admit an exact G2-structure. In a follow-up study, ansätze for geometric flow equations whose solutions may define closed G2-eigenforms are discussed.
Publications
- Homogeneous Spinor Flow, Q. J. Math. 71 (2020), no. 1, 21 – 51
M. Freibert, L. Schiemanowski, H. Weiß
(See online at https://doi.org/10.1093/qmathj/haz036) - Two-step solvable SKT shears (2020)
M. Freibert, A. Swann
- Closed G2 -eigenforms and exact G2-structures (2021)
M. Freibert, S. Salamon
