33 research outputs found
The spectral flow for Dirac operators on compact planar domains with local boundary conditions
Let , be an arbitrary 1-parameter family of Dirac type
operators on a two-dimensional disk with holes. Suppose that all
operators have the same symbol, and that is conjugate to by a
scalar gauge transformation. Suppose that all operators are considered
with the same locally elliptic boundary condition, given by a vector bundle
over the boundary. Our main result is a computation of the spectral flow for
such a family of operators. The answer is obtained up to multiplication by an
integer constant depending only on the number of the holes in the disk. This
constant is calculated explicitly for the case of the annulus ().Comment: 33 pages, 4 figures; section 9 adde
General Spectral Flow Formula for Fixed Maximal Domain
We consider a continuous curve of linear elliptic formally self-adjoint
differential operators of first order with smooth coefficients over a compact
Riemannian manifold with boundary together with a continuous curve of global
elliptic boundary value problems. We express the spectral flow of the resulting
continuous family of (unbounded) self-adjoint Fredholm operators in terms of
the Maslov index of two related curves of Lagrangian spaces. One curve is given
by the varying domains, the other by the Cauchy data spaces. We provide
rigorous definitions of the underlying concepts of spectral theory and
symplectic analysis and give a full (and surprisingly short) proof of our
General Spectral Flow Formula for the case of fixed maximal domain. As a side
result, we establish local stability of weak inner unique continuation property
(UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page
Weak UCP and perturbed monopole equations
We give a simple proof of weak Unique Continuation Property for perturbed
Dirac operators, using the Carleman inequality. We apply the result to a class
of perturbations of the Seiberg-Witten monopole equations that arise in Floer
theory.Comment: 22 pages LaTeX, one .eps figur
The Maslov index in weak symplectic functional analysis
We recall the Chernoff-Marsden definition of weak symplectic structure and
give a rigorous treatment of the functional analysis and geometry of weak
symplectic Banach spaces. We define the Maslov index of a continuous path of
Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces.
We derive basic properties of this Maslov index and emphasize the new features
appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom.
The final publication will be available at http://www.springerlink.com. arXiv
admin note: substantial text overlap with arXiv:math/040613
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
Modified differentials and basic cohomology for Riemannian foliations
We define a new version of the exterior derivative on the basic forms of a
Riemannian foliation to obtain a new form of basic cohomology that satisfies
Poincar\'e duality in the transversally orientable case. We use this twisted
basic cohomology to show relationships between curvature, tautness, and
vanishing of the basic Euler characteristic and basic signature.Comment: 20 pages, references added, minor corrections mad
Bifurcation of critical points along gap-continuous families of subspaces
We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations
Quantum gravity: unification of principles and interactions, and promises of spectral geometry
Quantum gravity was born as that branch of modern theoretical physics that
tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays
it is providing new insight into the unification of all fundamental interactions, while giving
rise to new developments in modern mathematics. It is however unclear whether it will ever
become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing
a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss
the general opinion that the mere mathematical complexity of the unification programme
will obstruct that programme