We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set of small surface area. These estimates involve either a volume bound or a spectral bound for the Dirac operator on a conformal compactification, but they are independent of the isoperimetric constant.
We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold $(M^n,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.
I construct multiplicies and orientations of tangent cones to any blow-up set $Z$ for the Seiberg-Witten equation with multiple spinors. This is used to prove that $Z$ determines a homology class, which is shown to be equal to the Poincaré dual of the first Chern class of the determinant line bundle. I also obtain a lower bound for the 1-dimensional Hausdorff measure of $Z$.
Motivated by Witten's spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.
We prove that a sequence of solutions of the Seiberg-Witten equation with multiple spinors in dimension three can degenerate only by converging (after rescaling) to a Fueter section of a bundle of moduli spaces of ASD instantons.
We consider a variant of the Seiberg-Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg-Witten equations in the setting of Kähler surfaces has a direct relation with ASD connections of holomorphic vector bundle. Also in Kähler setting, we construct a numerical invariant from the equations that detects a notion of $ϕ-$stability of $SU(n)-$holomorphic vector bundles where $ϕ$ is some prescribed non-trivial holomorphic section.
In this note, we present a proof of Donaldson's Diagonalization Theorem via an abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple spinors. Like the other proof of Donaldson's theorem using the standard Seiberg-Witten theory, Elkies' theorem also plays a key role in our argument.
I apply the algebraic classification of self-adjoint endomorphisms of ${\bf R}^{2,2}$ provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar results well known in four-dimensional Lorentzian geometry. The classification is summarized in Table 2 at the end of the paper.
This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds…
The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in $\mathcal{M}$ and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing $\mathcal{M}$ in terms of holomorphic data over the surface. We define analytic and algebro-geometric com…
Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.
In 1994, Witten has defined a monopole invariant and he has shown the equivalence of this invariant with Donaldson's polynomial using his result in \( \SS \)-duality. This new invariant is very powerful because the gauge group is abelian. By using such an invariant, many new results are found in the smooth, Kähler and even the symplectic categories. However, almost every paper in this topic write the monopole equations in a different way. Therefore is it necessary to clarify the basic idea behind the definition of such an invariant. In this paper we investigate the algebraic structure (Cliffor…
This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exoti…
In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.
Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.
We define (higher rank) spinorially twisted spin structures and deduce various curvature identites as well as estimates for the eigenvalues of the corresponding twisted Dirac operators.
We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the point-wise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand new solutions which do not come from the tensor product of Killing spinors and Killing-Yano forms.
We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.
Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.
I show that flat PSL(2;R)-connections on three-manifolds satisfying certain 'stability condition' can be interpreted as solutions of the Seiberg-Witten equations with two spinors. This is used to construct explicit examples of the Seiberg-Witten moduli spaces. Also, I show that in this setting blow up sets satisfy certain non-trivial topological restrictions.
We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with the Weitzenbock formula. The same curvature hypothesis implies the vanishing of the Seiberg-Witten invariant of any finite covering space. We also show that, in high dimensions, a spin manifold with mostly positive scalar curvature in fact admits a metric of positive scalar curvatu…
In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.
We study the geometric properties of a $2m$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $ξ$ up to scale. The defining property of $ξ$ is that it determines an almost null structure, ie an $m$-plane distribution $\mathcal{N}_ξ$ along which $g$ is …
Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. In fact, we show that the Lichnerowicz Laplacian, which governs the second variation, is the square of a twisted Dirac operator. Our second result, which is a local version of the first one, shows that any metrics of posi…
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.
We show that solutions of the Seiberg-Witten equations lead to non-trivial lower bounds for the L2-norm of the Weyl curvature of a compact Riemannian 4-manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics. These results considerably refine those previously obtained using scalar-curvature estimates alone.
We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids).
We study the geometric properties of a $(2m+1)$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m+1,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $ξ$ up to scale. The defining property of $ξ$ is that it determines an almost null structure, i.e.\ an $m$-plane distribution $\mathcal{N}_ξ$ along whic…
We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results f…
In this article, we give all the Weitzenböck-type formulas among the geometric first order differential operators on the spinor fields with spin $j+1/2$ over Riemannian spin manifolds of constant curvature. Then we find an explicit factorization formula of the Laplace operator raised to the power $j+1$ and understand how the spinor fields with spin $j+1/2$ are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of…
Let $(\overline M,\overline g)$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $\overline M$ with induced Riemannian metric $g$ and second fundamental form $K$. If $(\overline M,\overline g)$ satisfies the dominant energy condition in a strict sense, then the Dirac--Witten operator of $M\subseteq \overline M$ is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on $M$ satisfying the dominant energy condition in a strict …
(This is a revised version of the paper) - In the present paper we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariantmetric of Lorentzian signature $(1,n-1)$, of signature $(2,n-2)$ and of signature $(p,q)$, where $p+q\leq 6$. Furthermore, some special series with higher signature are discussed.
This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Clifford module is a direct sum of C2 bundles associated to SpinC structures; and the case of the SU(2) Vafa-Witten equations with an Abelian ansatz.
Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O. Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to $3$ is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian Spin manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.
In this paper, we write down Seiberg-Witten equations on contact metric manifolds of dimension 5. Any contact metric manifold has a spin^c structure. For Dirac equation we use Dirac type operators associated to the generalized Tanaka-Webster connection on spin^c spinor bundle of a contact metric manifold. For curvature equation we need to self-duality concept. Self-duality concept is significant on odd dimensional manifolds, particularly, on 5-dimensional contact manifolds. Finally, we give a global solution to these equations on strictly pseudoconvex CR manifolds.
We consider Riemannian 4-manifolds $(X,g_X)$ with a Spin^c-structure and a suitable circle bundle $Y$ over $X$ such that the Spin^c-structure on $X$ lifts to a spin structure on $Y$. With respect to these structures a spinor $ϕ$ on $X$ lifts to an untwisted spinor $ψ$ on $Y$ and a U(1)-gauge field $A$ for the Spin^c-structure can be absorbed into a Kaluza-Klein metric $g_Y^A$ on $Y$. We show that irreducible solutions $(A,ϕ)$ to the Seiberg-Witten equations on $(X,g_X)$ for the given Spin^c-structure are equivalent to irreducible solutions $ψ$ of a Dirac equation with cubic non-linearity on th…
We study symplectic manifolds $(M^{2l},ω)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $Γ(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the m…
Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\…
We define functionals generalising the Seiberg-Witten functional on closed $spin^c$ manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then prove energy estimates along the flow, and establish local $L^2$-derivative estimates. These are then used to show long time existence of the flow in sub-critical dimensions. In the critical dimension, we are able to show that long time existence is obstructed by an $L^{k+2}$ curva…
The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect the sign of the answer.) In this article, it is shown that many 4-manifolds M with Y(M) < 0 have have finite covering spaces \tilde{M} with Y(\tilde{M}) > 0.
We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta, we compute these invariants in many cases that were previously intractable. In particular, we are now able to calculate the Yamabe invariant for certain connected sums of complex surfaces.
We study the space of positive scalar curvature (psc) metrics on a 4-manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1-parameter Seiberg-Witten invariants which we introduced in earlier work. The invariant shows that the diffeomorphism group of the underlying 4-manifold is disconnected. We also study the moduli space of positive scalar curvature metrics modulo diffeomorphism, and give examples to show that this spa…
Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University（湖北大学数学与统计学学院，湖北省应用数学重点实验室）；Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education（教育部智能感知系统与安全重点实验室（湖北大学））
Since the first work of Thomas Friedrich showing that isometric immersions of Riemann surfaces are related to spinors and the Dirac equation, various works appeared generalizing this approach to more general Spin-manifolds, in particular the case of submanifolds of Spin-manifolds of constant curvature. In the present work we investigate the case of submanifolds of SpinC-manifolds of constant curvature.
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler structures do not admit holomorphic vector fields. Following Witten, we define a complex variant of the Hitchin connection on the bundle of prequantum spaces. The curvature is essentially unchanged, so projective flatness holds in the same cases. Finally, the results are applied to qua…
In this article, we study a generalisation of the Seiberg-Witten equations, replacing the spinor representation with a hyperKahler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKahler manifolds admitting a hyperKahler potential, we derive a transformation formula for the Dirac operator under the conformal change of metric on the base manifold. As an application, we show that when the hyperKahler manifold is of dimension four, then away from a singular set, the equat…
Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariant vanishes for all \spinc structures.
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=Δ_ϕ(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such en…
A closed spin Kähler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin Kähler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing theorems for holomorphic spinors are proved.
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmoni…
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold. For a class of hyperKahler manifolds obtained via hyperKahler reduction, we also show that a harmonic spinor, under mild conditions, defines a symplectic structure. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized Seiberg-Witten equati…
