33 research outputs found

    The spectral flow for Dirac operators on compact planar domains with local boundary conditions

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    Let DtD_t, t[0,1]t \in [0,1] be an arbitrary 1-parameter family of Dirac type operators on a two-dimensional disk with m1m-1 holes. Suppose that all operators DtD_t have the same symbol, and that D1D_1 is conjugate to D0D_0 by a scalar gauge transformation. Suppose that all operators DtD_t 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 (m=2m=2).Comment: 33 pages, 4 figures; section 9 adde

    General Spectral Flow Formula for Fixed Maximal Domain

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    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

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    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

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    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

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    We show that in the analytic category, given a Riemannian metric gg on a hypersurface MZM\subset \Z and a symmetric tensor WW on MM, the metric gg can be locally extended to a Riemannian Einstein metric on ZZ with second fundamental form WW, provided that gg and WW satisfy the constraints on MM 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

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    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

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    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

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    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
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