The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of quaternionic holomorphic 2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4). After describing how the Sato-Segal-Wilson construction of KP solutions (particularly solutions of finite type) carries over to this quaternionic setting, we compare three different notions of "spectral curve": the QKP spectral curve, which arises from an algebra of commuting differential operators; the (unnormalised) Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The latter two are shown to be images of the QKP spectral curve, which need not be smooth. Moreover, it is a singularisation of this QKP spectral curve, rather than the normalised Floquet multiplier curve, which determines the classification of conformally immersed 2-tori of finite spectral genus.
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Ampère equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.
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…
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a study is two-fold: it comes from the constantly growing interest in Weyl (and Einstein-Weyl) geometry and, on the other hand, from the necessity of understanding the existing classes of quaternion Hermitian manifolds.
We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-Riemannian manifolds admitting conformal Killing fields satisfying a conformally invariant system of conditions analog to G. Sparling's criteria; and as the total space of a SO(3)- or $S^3$-bundle over a qc manifold with the conformally equivalent metrics defined directly by Biqu…
In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.
We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally Kähler.
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…
This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, …
We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $γ$ in such a torus is said to be homologically maximizing if one (hence every) lift of $γ$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics. This yields the Lipschitz continuity of the time separation of the universal cover on strict sub-cones of the cone of future pointing vectors. Then we introduce the stable time separation $\mathfrak{l}$. As an application we prove relations between the concavity properties of $…
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 study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwise.
Penrose's two-spinor notation for $4$-dimensional Lorentzian manifolds can be extended to two-component notation for quaternionic manifolds, which is a very useful tool for calculation. We construct a family of quaternionic complexes over unimodular quaternionic manifolds by elementary calculation. On complex quaternionic manifolds, which are essentially the complexification of real analytic quaternionic Kähler manifolds, the existence of these complexes was established by Baston by using twistor transformations and spectral sequences. Unimodular quaternionic manifolds constitute a nice and la…
The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori which is contained in a hypersphere in the complex Euclidean plane are constructed explicitly. Then it is shown that they are homogeneous tori. For the construction, flat quaternionic connections of Hamiltonian stationary Lagrangian tori are considered and a spectral curve of an asso…
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results…
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 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.
Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting examples of constrained Willmore tori are the tori with constant mean curvature (CMC) in a 3-dimensional space form. It is shown in [14] and in [16] that the spectral curves of these tori are hyperelliptic. In this paper we show under mild conditions that a constrained Willmore torus…
Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit …
This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in $\R^4$. This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geometry, since the group of ambient isometries acts non-trivially on the spectral data and the relevant energy functional (the area) need not be constant under deformations by higher flows.
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 …
A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8π$. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution …
This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometr…
