Compact Spin Manifold Research Articles

Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

Abstract In this article, we study the existence of solutions for the Dirac system { D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v ) on ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v ) on ⁢ M , \left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right. where M is an m-dimensional compact Riemannian spin manifold, u , v ∈ C ∞ ⁢ ( M , Σ ⁢ M ) {u,v\in C^{\infty}(M,\Sigma M)} are spinors, D is the Dirac operator on M, and the fiber preserving map H : Σ ⁢ M ⊕ Σ ⁢ M → ℝ {H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}} is a real-valued superquadratic function of class C 1 {C^{1}} with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

Existence of infinitely many solutions of Dirac equations with sublinear nonlinearity

In this note we use an recent improvement of Clark’s theorem by Liu and Wang (Nonlinear Anal 32:1015–1037, 2015) to prove two existence results of infinitely many solutions for sublinear Dirac equations and systems on compact spin manifolds.

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$-orientation from the infinite loop space of the Madsen--Tillmann--Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$-dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov--Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah--Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral-flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen--Weiss theorem, proven by Galatius and the third named author.

On coupled Dirac systems

In this paper, we show the existence of solutions for the coupled Dirac system \begin{document}$\left\{ \begin{aligned}Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\\Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\end{aligned} \right.$\end{document} where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the $\mathbb{Z}_2$-invariant $H$ that includes a nonlinearity of the form \begin{document}$H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1},$\end{document} where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy \begin{document}$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}.$\end{document} In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.

Morse–Floer theory for superquadratic Dirac equations, II: construction and computation of Morse–Floer homology

We give a construction and computation of Morse–Floer homology for a class of superquadratic Dirac functionals defined on the set of spinors on a compact spin manifold. For the superquadratic functionals, we show that the Morse–Floer homology is well defined for generic choice of metric on the set of spinors and its isomorphism class is independent of the choice of such a generic metric. Moreover, we show that it is also independent of the choice of superquadratic Dirac functional. Therefore, it defines an invariant for the set of spinors which we call the superquadratic-Dirac–Morse–Floer homology. We prove a vanishing result for this homology. As an application, we give existence and multiplicity results for a class of superquadratic Dirac equations on compact spin manifolds.

Morse–Floer theory for superquadratic Dirac equations, I: relative Morse indices and compactness

In this paper and its sequel (Isobe in Morse–Floer theory for superquadratic Dirac equations, II: construction and computation of Morse–Floer homology, 2016), we study Morse–Floer theory for superquadratic Dirac functionals associated with a class of nonlinear Dirac equations on compact spin manifolds. We are interested in two topics: (i) relative Morse indices and its relation to compactness issues of critical points; (ii) construction and computation of the Morse–Floer homology and its application to the existence problem for solutions to nonlinear Dirac equations. In this part I, we focus on the topic (i). One of our main results is a compactness of critical points under the boundedness assumption of their relative Morse indices which is an analogue of the results of Bahri–Lions (Commun Pure Appl Math 45:1205–1215, 1992) and Angenent–van der Vorst (Math Z 231: 203–248, 1999) for Dirac functionals. To prove this, we give an appropriate definition of relative Morse indices for bounded solutions to \(\mathsf {D}_{g_{\mathbb {R}^{m}}}\psi =|\psi |^{p-1}\psi \) on \(\mathbb {R}^{m}\). We show that for \(m\ge 3\) and \(1<p<\frac{m+1}{m-1}\), the relative Morse index of any non-trivial bounded solution to that equation is \(+\infty \). We also give some useful properties of the relative Morse indices of Dirac functionals which will be used in the study of the topic (ii) above.

Solutions of Dirac equations on compact spin manifolds via saddle point reduction

In this note, we apply the saddle point reduction to a class of nonlinear Dirac equation with a potential on a compact spin manifold. Two results are obtained, including a multiple theorem.

Bifurcation on compact spin manifold

We consider the bifurcation phenomenon of solutions to a class of nonlinear Dirac equations \(\mu D\psi =\psi +h(\psi )\) on compact spin manifolds. There are two main results. The first is bifurcation from zero, which asserts that the equations allow a sequence of bifurcation points \((\mu _{k},\theta )\), and for each \(\mu \in \mathbb {R}\) near \(\mu _{k}\) the corresponding equation has at least two nontrivial solutions near \(\theta \). The second result is bifurcation from infinity. The point \((\mu _{k},\infty )\) is also proved to be a bifurcation point and we give a characterization of the bifurcation behavior.

Existence of Dirac eigenvalues of higher multiplicity

In this article, we prove that on any compact spin manifold of dimension m congruent 0,6,7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by catching the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.

Spin-structures and proper group actions

We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for an Aˆ-type genus.

A spinorial energy functional: critical points and gradient flow

Let M be a compact spin manifold. On the universal bundle of unit spinors we study a natural energy functional whose critical points, if $$\dim M \ge 3$$ , are precisely the pairs $$(g,{\varphi })$$ consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor $${\varphi }$$ . We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

Finite part of operatorK–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds

In this paper, we study lower bounds on the K‐theory of the maximal C ‐algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K‐theory and give a lower bound that is valid for a large class of groups, called the finitely embeddable groups. The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov’s monster groups, virtually torsion-free groups (eg Out.Fn/), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. We apply this result to measure the degree of nonrigidity for any compact oriented manifold M with dimension 4k 1 .k > 1/. In this case, we derive a lower bound on the rank of the structure group S.M/, which is roughly defined to be the abelian group of all pairs .M 0 ;f /, where M 0 is a compact manifold and fW M 0 ! M is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group z S.M/, which measures the size of the collections of compact manifolds homotopic equivalent to but not homeomorphic to M by any homeomorphism at all (not necessary homeomorphism in the homotopy equivalence class). For a compact Riemannian manifold M with dimension greater than or equal to 5 and positive scalar curvature metric, there is an abelian group P.M/ that measures the size of the space of all positive scalar curvature metrics on M . We obtain a lower bound on the rank of the abelian group P.M/ when the compact smooth spin manifold M has dimension 2k 1 .k > 2/ and the fundamental group of M is finitely embeddable. 19K99; 20F99, 58D29

On Dirac equation with a potential and critical Sobolev exponent

In this paper we consider a critical Dirac equation with a potential on a compact spin manifold. We prove the existence of solutions based on the analysis of the spectrum of Dirac operator with a potential and the dual variational method.

On the space of connections having non-trivial twisted harmonic spinors

We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yields an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.

A NOTE ON GENERALIZED DIRAC EIGENVALUES FOR SPLIT HOLONOMY AND TORSION

We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection <TEX>${\nabla}$</TEX> with skew torsion <TEX>$T{\in}{\Lambda}^3M$</TEX> in the situation where the tangent bundle splits under the holonomy of <TEX>${\nabla}$</TEX> and the torsion of <TEX>${\nabla}$</TEX> is of 'split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.

Rho-classes, index theory and Stolz’ positive scalar curvature sequence

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group G, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to the surgery exact sequence in topology. It calculates a structure group of metrics of positive scalar curvature on M (the object we want to understand) in terms of spin-bordism of BG and a somewhat mysterious group R(G). Higson and Roe introduced a K-theory exact sequence in coarse geometry which contains the Baum-Connes assembly map, with one crucial term K(D*G) canonically associated to G. The K-theory groups in question are the home of interesting index invariants and secondary invariants, in particular the rho-class in K_*(D*G) of a metric of positive scalar curvature on a spin manifold. One of our main results is the construction of a map from the Stolz exact sequence to the Higson-Roe exact sequence (commuting with all arrows), using coarse index theory throughout. Our main tool are two index theorems, which we believe to be of independent interest. The first is an index theorem of Atiyah-Patodi-Singer type. Here, assume that Y is a compact spin manifold with boundary, with a Riemannian metric g which is of positive scalar curvature when restricted to the boundary (and with fundamental group G). Because the Dirac operator on the boundary is invertible, one constructs a delocalized APS-index in K_* (D*G). We then show that this class equals the rho-class of the boundary. The second theorem equates a partitioned manifold rho-class of a positive scalar curvature metric to the rho-class of the partitioning hypersurface.

Twistorial eigenvalue estimates for generalized Dirac operators with torsion

We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ∇ with skew torsion T∈Λ3M by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich’s classical Riemannian estimate. We also determine a novel twistor and the Killing equation with torsion and use it to discuss the case in which the minimum is attained in the bound.

Electrodynamics from noncommutative geometry

Within the framework of Connes’ noncommutative geometry, the notion of an almost commutative manifold can be used to describe field theories on compact Riemannian spin manifolds. The most notable example is the derivation of the Standard Model of high energy physics from a suitably chosen almost commutative manifold. In contrast to such a non-abelian gauge theory, it has long been thought impossible to describe an abelian gauge theory within this framework. The purpose of this paper is to improve on this point. We provide a simple example of a commutative spectral triple based on the two-point space and show that it yields a U(1)-gauge theory. Then we slightly modify the spectral triple such that we obtain the full classical theory of electrodynamics on a curved background manifold.

Bergman and Calderón Projectors for Dirac Operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$ , we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L 2 harmonic spinors on $\bar{X}$ , and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$ . We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

On gravity, torsion and the spectral action principle

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine–Connes spectral action, deduce the equations of motions and discuss critical points.