A Z2-harmonic spinor on a 3-manifold Y is a solution of the Dirac equation on a bundle that is twisted around a submanifold Z of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of Z2-harmonic spinors over the space of parameters (g, B) consisting of a metric and perturbation to the spin connection. The main result states that near a Z2-harmonic spinor with Z smooth, the universal moduli space projects to a co-dimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set Z, necessitating the use of the Nash-Moser Implicit Function Theorem.
The half-line Dirac operators with L^2 -potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general L^2 -case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with δ -interactions on a half-lattice in terms of the Schur’s algorithm for analytic functions.
We prove existence results for optimization problems for the k th Laplace eigenvalue on closed Riemannian manifolds of dimension m ≥ 3 , depending on the choice of normalization. One such normalization leads to eigenvalue optimization within a conformal class, for which existence of maximizers was previously known only in dimension two. We also prove that all absolutely continuous maximizers of the normalized eigenvalue functionals are always induced by p -harmonic maps into spheres, where p ∈ [2,m] . For p sufficiently close to m , the maximizers are Hölder-continuous, whereas for p< m no bubbling occurs. A key tool in our analysis is the application of techniques from the theory of topological tensor products, which appear to be well suited for studying eigenvalue-related optimization problems.
A dynamical system (X,T) is shift embeddable if (X,T) embeds continuously and equivariantly in the shift over [0,1]^d for some finite d . Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov’s mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embeddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.
