Let M be a compact manifold with a spin structure χand a Riemannian metric g. Let λ_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M,χ):= sup inf \sqrt{λ_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that τ(T^2,χ)=2\sqrtπ if χis ``the'' non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2\sqrtπ at one end of the spin-conformal moduli space.
We show how to assign to any immersed torus in $\R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the torus and, in particular, relates to the Willmore functional. We propose a simple proof that for isothermic tori in $\R^3$ (this class includes constant mean curvature tori and tori of revolution) the spectrum is invariant with respect to conformal transformations of $\R^3$. We show …
This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $λ_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many…
We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so far that uses information about the spin structure. As a corollary we obtain lower bounds for the Willmore functional of a torus embedded into S_3. In the final section we compare Dirac spectra for two different spin structures on an arbitrary Riemannian spin manifold.
For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.
We consider conformal immersions $f: T^2\rightarrow \mathbb{R}^3$ with the property that $H^2 f^*g_{\mathbb{R}^3}$ is a flat metric. These so called Dirac tori have the property that its Willmore energy is uniformly distributed over the surface and can be obtained using spin transformations of the plane by eigenvectors of the standard Dirac operator for a fixed eigenvalue. We classify Dirac tori and determine the conformal classes realized by them. We want to note that the spinors of Dirac tori satisfies the same system of PDE's as the differential of Hamiltonian stationary Lagrangian tori in …
We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fixed conformally congruent area. On the way to establish this result we prove that tori which are critical points of the area for perturbations within a given conformal class and which are degenerate points of the conformal class mapping - i.e. isothermic - are either minimal surf…
In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.
In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in $C^1$. The full conjecture from the conformal perspective is also discu…
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean c…
For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4πk$. Moreover, we show that the energy value $8π$ cannot be attained by such an immersion. Additionally, we characterize the branched double covers $T^2\to S^2 \times \{0\}$ as the only branched conformal immersions, up to Möbius transformations of $\mathbb{R}^4$, from a torus into $\mathbb{R}^4$ with at least one branch point and Willmore energy $8π$. Using a perturbation argument in order t…
We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.
We give results about the L^2 kernel and the spectrum of the Dirac operator on a complete Riemannian manifold which is conformally equivalent to the interior of a Riemannian manifold with nonempty boundary.
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 obtain two kinds of Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of signature operators by a vector bundle with a non-unitary connection on six-dimensional manifolds with (respectively without)boundary.
This paper constructs a family of conformally invariant differential operators acting on spinor densities with leading part a power of the Dirac operator. The construction applies for all powers in odd dimensions, and only for finitely many powers in even dimensions. These operators arise naturally as obstructions to formal solution of the Dirac equation on the Fefferman-Graham ambient space with prescribed boundary conditions.
We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be the unique constrained Willmore minimizers (up to invariance). In this paper we generalize this result and show that the candidates constructed in \cite{HelNdi2} are indeed constrained Willmore minimizers in certain non-rectangular conformal classes $\;(a,b).\;$ Difficulties arise from the fact that these minimizers are non-degenerate for $\;a \neq 0\;$ but smoothly converge…
We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in the 4-sphere can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin's approach to the study of harmonic tori in the 3-sphere.
We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R^3. This hypothesis was formulated recently by I.A.Taimanov (dg-ga/9610013). Higher analogs of the Willmore functional are defined in terms of the Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with the zero-energy scattering problem for the two-dimensional Dirac operator.
In this paper we consider the coupled system given by the first variation of the conformal Dirac-Einstein functional. We will show existence of solutions by means of perturbation methods.
The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in terms of associated tractor bundles to compute these correction terms. Depending on the signature of the semi Riemannian manifold in question, the obtained conformal powers of the Dirac operator turn out to be formally self-adjoint with respect to the $L^2-$scalar product, or formall…
We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.
In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elastic curve with enclosed area constraint, in the round 2-sphere under the Hopf fibration. We show that all conformal types can be isometrically immersed into S^3 as constrained Willmore (Hopf) tori and write down all constrained elastic curves in H^2 and S^2 in terms of the Weiers…
The Courant bracket defined originally on the sections of the vector bundle $TM \oplus T^*M \to M$ is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows to interpret many structures encountered in differential geometry in terms of Dirac structures. We give here a new approach to conformal Jacobi structures.
In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries.
We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second term of the short time heat kernel expansion for the perturbed Laplacians on functions and on 1-forms. The Ricci curvature is defined by localizing heat traces suitably. Equiv…
The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of the left normal, as defined by Hitchin. We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces o…
We study the action of conformal transformations of the ambient space on the Dirac operator coming into the Weierstrass (or spinor) representation of a torus in the Euclidean four-space. It is showed that such an action generates a flow acting on the potential of the operator, that this flow is described by a nonlinear system of the Melnikov type and that it preserves the Floquet multipliers of the Dirac operator with double-periodic potential. However this flow is only almost isospectral since it does not preserve the spectral curve in general and its action may result in adding or removing m…
In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with flat boundary. In particular, for 6-dimensional spin manifolds with boundary with the conformal Robertson-Walker metric, we obtain the noncommutative residue of the composition of $π^+D^{-1}$ and $π^+D^{-3}$ is proportional to the Einstein-Hilbert action for manifolds with bound…
In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result.
In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of …
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the associated Poincaré-Einstein metric, and relies on combinatorial recurrence identities related to the dual Hahn polynomials.
Let us fix a conformal class $[g_0]$ and a spin structure $σ$ on a compact manifold $M$. For any $g\in [g_0]$, let $λ^+_1(g)$ be the smallest positive eigenvalue of the Dirac operator $D$ on $(M,g,σ)$. In a previous paper we have shown that $$λ_{min}(M,g_0,σ):=\inf_{g\in [g_0]} λ_1^+(g)\vol(M,g)^{1/n}>0.$$ In the present article, we enlarge the conformal class by certain singular metrics. We will show that if $λ_{min}(M,g_0,σ)<λ_{min}(S^n)$, then the infimum is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial differential equations in…
The Moutard transformation for a two-dimensional Dirac operator with a complex-valued potential is constructed. It is showed that this transformation relates the potentials of Weierstrass representations of surfaces related by a composition of the inversion and a reflection with respect to an axis. It is given an analytical description of an explicit example of such a transformation which results in a creation of double points on the spectral curve of a Dirac operator with a double-periodic potential.
We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,α}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its …
We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_θ^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_θ^2$.
We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying…
In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds treated here include for example $RP^{n}$ and $S^{1}\times S^{n-1}$. Special kinds of Clifford-analytic automorphic forms associated to the different choices of $Γ$ are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spa…
It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It seems to be a folklore fact that the metric can be reconstructed up to conformal equivalence if one replaces the Dirac operator D by sign(D). We give a precise formulation and proof of this fact.
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.
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly Kähler compact manifolds.
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.
It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum gravity models by quantizing the moduli of Laplace, weight and defining-function operators on Fefferman-Graham ambient spaces. The algebra of these operators underlies conformal geometries. We extend those results to include fermions by taking an osp(1|2) "Dirac square root" of these…
For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of 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 to the complete conformal metric $g=\bar{g}/ρ^2$ where $ρ$ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$. We show that $({\rm I…
