Browsing by Subject "Geometry"

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

To every Legendrian link in R3, we can assign a differential graded algebra (DGA) called the Chekanov-Eliashberg DGA. An exact Lagrangian cobordism between two Legendrian links induces a DGA map between the corresponding Chekanov-Eliashberg DGAs, and this association is functorial. This DGA map was written down explicity for exact, decomposable Lagrangian fillings as Z_2-count of certain pseudoholomorphic disks by Ekholm, Honda, and K ́alm ́an, and this was combinatorially upgraded to an integral count by Casals and Ng. However, this upgrade only assigned an automorphism class of DGA maps. We approach the same problem of integral lifts by a different strategy, first done for the differential in the Chekanov-Eliashberg DGA by Ekholm, Etnyre, and Sullivan. Here, we find the precise DGA maps for all exact, decomposable Lagrangian cobordisms through this more analytic method.

This dissertation examines two different subjects within the study of instantons: the construction of Nahm transforms for instantons invariant under certain group actions; and a generalization of the proof that Yang--Mills minimizers are instantons.

The first Nahm transform examined is the ADHM construction for $S^1$-invariant instantons on $S^4$, which correspond to singular monopoles on $\RR^3$. In this case, there is a decomposition of the ADHM data in terms of $S^1$-subrepresentations of $\ker \Dir$. The moduli spaces of $S^1$-invariant $SU(2)$-instantons are given up to charge 3, and examples of ADHM data for instantons of charge $4$ are also provided.

The second Nahm transform considered is for instantons on a certain flat quotient of $\RR^4$ with nonabelian fundamental group. Equivalently, one can consider these to be $\ZZ_2$-invariant instantons on $T^4$, and the Nahm transform yields instantons invariant under a crystallographic action.

In our study of minimal Yang--Mills connections, we extend results of Bourguignon--Lawson--Simons and Stern, who showed that connections that minimize $\|F_\nabla\|^2$ on homogeneous manifolds must be instantons or have instanton subbundles. We extend the previous arguments by considering variations constructed using conformal vector fields, and also allow these vector fields to be incomplete. We prove a minimality result over a half-cylinder.