Compact Spin Manifold Research Articles

Nonlinear Dirac Equation on Compact Spin Manifold with Chirality Boundary Condition

Nonlinear Dirac Equation on Compact Spin Manifold with Chirality Boundary Condition

Multiple solutions to the nonlinear Dirac systems on compact spin manifolds with boundary

Multiple solutions to the nonlinear Dirac systems on compact spin manifolds with boundary

A short proof of the localization formula for the loop space Chern character of spin manifolds

In this note, we give a short proof of the localization formula for the loop space Chern character of a compact Riemannian spin manifold M , using the rescaled spinor bundle on the tangent groupoid associated to M .

Secondary higher invariants and cyclic cohomology for groups of polynomial growth

We prove that if \Gamma is a group of polynomial growth, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocycle, we thus define a higher analogue of Lott’s delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if \Gamma is of polynomial growth, then there is a well-defined pairing between delocalized cyclic cocycles and K -theory classes of C^* -algebraic secondary higher invariants. When this K -theory class is that of a higher rho invariant of an invertible differential operator, we show this pairing is precisely the aforementioned higher analogue of Lott’s delocalized eta invariant. As an application of this equivalence, we provide a delocalized higher Atiyah–Patodi–Singer index theorem, given that M is a compact spin manifold with boundary, equipped with a positive scalar metric g and having fundamental group \Gamma=\pi_1(M) which is finitely generated and of polynomial growth.

Eigenvalues of the Dirac operator on compact spin manifolds under Ricci flow

In this paper, we first derive the evolution formula for the square of the first eigenvalue λ2 of the Dirac operator under the metric flow ∂∂tg=h on compact spin manifolds. We then prove that λ2 is nondecreasing under the Ricci flow on compact spin surfaces with non-negative Gauss curvature.

First Boundary Dirac Eigenvalue and Boundary Capacity Potential

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface \(\Sigma \) bounding a noncompact domain in a spin asymptotically flat manifold \((M^n,g)\) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of \(\Sigma \) in \((M^n,g)\) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Equivariant Toeplitz index theory on odd-dimensional manifolds with boundary

In this paper, we establish an equivariant version of Dai-Zhang's Toeplitz index theorem for compact odd-dimensional spin manifolds with even-dimensional boundary.

A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

We give a rigorous construction of the path integral in {mathcal {N}}=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten.

Global Strichartz estimates for the Dirac equation on symmetric spaces

Abstract In this paper, we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq 3$ . In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m≥2, let S(M) denote the spinor bundle on M, and let D be the Atiyah-Singer Dirac operator acting on spinors ψ:M→S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m−1ψ where λ∈R and f(|ψ|)ψ is a subcritical nonlinearity in the sense that f(s)=o(s2m−1) as s→∞. A model nonlinearity is f(s)=αsp−2 with 2<p<2mm−1, α∈R. In particular we study the nonlinear Dirac equation(BND)Dψ=λψ+|ψ|2m−1ψ,λ∈R. This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (λ,ψ) of (BND) for every λ>0, even if λ is an eigenvalue of D. For some classes of nonlinearities f we also obtain solutions of (NLD) for every λ∈R, except for non-positive eigenvalues. If m≢3 (mod 4) we obtain solutions of (BND) for every λ∈R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it.The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).

An invariant related to the existence of conformally compact Einstein fillings

We define an invariant for compact spin manifolds X X of dimension 4 k 4k equipped with a metric h h of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of h h to be the conformal infinity of a conformally compact Einstein metric on X X .

A long neck principle for Riemannian spin manifolds with positive scalar curvature

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if {{,mathrm{scal},}}(X)ge n(n-1) and there is a nonzero degree map into the sphere f:Xrightarrow S^n which is strictly area decreasing, then the distance between the support of text {d}f and the boundary of X is at most pi /n. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if {{,mathrm{scal},}}(X)>sigma >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of partial X is at most pi sqrt{(n-1)/(nsigma )}. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to Ntimes [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if {{,mathrm{scal},}}(V)ge sigma >0, then the distance between the boundary components of V is at most 2pi sqrt{(n-1)/(nsigma )}. This last constant is sharp by an argument due to Gromov.

Positive scalar curvature metrics via end-periodic manifolds

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka-Ruberman-Saveliev (MRS). These results are the even dimensional analogs of the results by Higson-Roe. The second type of result studies the number of path components of the space of positive scalar curvature metrics modulo diffeomorphism for compact spin manifolds that are even dimensional, whenever this space is non-empty. These extend and refine certain results in Botvinnik-Gilkey and also MRS. End-periodic analogs of K-homology and bordism theory are defined and are utilised to prove many of our results.

Generalized linking theorem with applications to nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.

Sharp decay estimates for critical Dirac equations

We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.

Comparison between two differential graded algebras in noncommutative geometry

Starting with a spectral triple, one can associate two canonical differential graded algebras (DGA) defined by Connes (Noncommutative geometry (1994) Academic Press Inc., San Diego) and Frohlich et al. (Comm. Math. Phys. 203(1) (1999) 119–184). For the classical spectral triples associated with compact Riemannian spin manifolds, both these DGAs coincide with the de-Rham DGA. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two DGAs in a very precise sense.

The Kernel of the Rarita–Schwinger Operator on Riemannian Spin Manifolds

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion K\"ahler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.

Subadditivity and additivity of the noncommutative Yang-Mills action functional

We formulate notions of subadditivity and additivity of the Yang-Mills action functional in noncommutative geometry. We identify a suitable hypothesis on spectral triples which proves that the Yang-Mills functional is always subadditive. The additivity property is stronger in the sense that it implies the subadditivity property. Under our hypothesis, we also obtain a necessary and sufficient condition for the additivity of the Yang-Mills functional. An instance of additivity is shown for the case of noncommutative n-tori. We also investigate the behavior of critical points of the Yang-Mills functional under additivity. In the end, we discuss examples involving compact spin manifolds, matrix algebras, noncommutative n-torus, and the quantum Heisenberg manifolds to validate our hypothesis.

A boundary value problem for the nonlinear Dirac equation on compact spin manifold

The positive energy theorem is a significant subject in general relativity theory. In Witten’s proof of this theorem, the solution of a free Dirac equation which is a spinor filed plays an important role. In order to prove the positive energy theorem for black holes, Gibbons, Hawking, Horowitz and Perry imposed a local boundary condition on the apparent horizon of the black hole. Then the Dirac equation under this boundary condition forms an elliptic boundary value problem. In fact, this kind of local boundary condition can be generally defined by a Chirality operator on the Dirac bundle over a spin manifold. In this paper, by establishing a proper analysis setting and developing variational arguments, we study a nonlinear Dirac equation on a compact spin manifold (M, g) which satisfies the local boundary condition with respect to a Chirality operator.