We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-space Y, with X = Y/G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.
We study the transversely metaplectic structure and the transversely symplectic Dirac operator on a transversely symplectic foliation. Moreover, we give the Weitzenbock type formula for transversely symplectic Dirac operators and we estimate the lower bound of the eigenvalues of the transversely symplectic Dirac operator on special spinors space on transverse Kahler foliations.
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with (resp.without) boundary
In this study, we obtain a spinorial Gauss formula for a lightlike hypersurface in Lorentzian manifold with 4-dimension. Then, we take into account the changes caused by degenerate metric on hypersurface and investigate Dirac operator for lightlike hypersurface. Later, we establish the relation between Dirac operators and Riemannian curvatures of manifold and hypersurface.
In this paper, we prove two Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of twisted signature operators on four-dimensional manifolds with (resp. without) boundary.
In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.
Given a closed connected spin manifold M with non-negative and somewhere positive scalar curvature, we show that the Dirac operator twisted with any flat Hilbert module bundle is invertible.
We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction.
We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.
