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 of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.
We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more trad…
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…
Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one
We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $\mathbb{R}^{2n}$.
We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kähler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kähler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kähler metric w…
In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangian torus orbits of compact toric Kahler manifolds.
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.
Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They can be considered as a generalization of special Lagrangians or Lagrangian and minimal submanifolds. Joyce, Schoen and the author show that given any compact rigid Hamiltonian stationary Lagrangian in $\C^n$, one can always find a family of Hamiltonian stationary Lagrangians of the same type in any compact symplectic manifolds with a compatible metric. The advantage of this result is that it holds in very general classes. But the disadvantage is …
In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows starting from these examples. We allow these flows to have some singularities and topological changes.
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.
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).
In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
We study geometric properties of the Lagrangian self-shrinking tori in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in \cite{CMa}. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is com…
We study the energy functional on the set of Lagrangian tori in $\mathbb{C}P^2$ . We prove that the value of the energy functional on a certain family of Hamiltonian minimal Lagrangian tori in $\mathbb{C}P^2$ is strictly larger than energy of the Clifford torus.
We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.
In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
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 show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.
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.
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with negative first Chern class.
In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric variety near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit $X_0$. In this paper, we will deal with the region near the smooth top dimensional torus fibres of $X_0$ and its mirror dual situation: the region near the 0-dimensional fibres of $…
A Hamiltonian stationary Lagrangian submanifold of a Kaehler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kaehler manifold of real dimension four that guarantees the existence of a family of small Hamiltonian stationary Lagrangian tori.
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is well known. What does not appear to be in the current literature is an explicit description of such examples in which the special Lagrangian submanifold is a 3-torus.
We show the existence of special Lagrangian tori on one family of Borcea-Voisin threefolds. We also construct a family of special Lagrangian submanifolds on the total space of the canonical line bundle of projective spaces.
In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schrödinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the potential of the Schrödinger operators, which has a natural geometrical meaning. We study the energy functional on two families of Lagrangian tori and propose a conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori. In particular we show that if the deformation preserves a conformal…
We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible unitary connections supported on complex submanifolds of the relative Jacobian of $X$ (suitable conditions being verified on both sides).
We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries. We then recover a result of Morgan-Mrowka-Szabó on product formulas for the Seiberg-Witten invariants along 3-tori.
We prove a Thomas--Yau-type conjecture for monotone Lagrangian tori satisfying a symmetry condition in the complex projective plane $\mathbb{CP}^2$. We show that such tori exist for all time under Lagrangian mean curvature flow with surgery, undergoing at most a finite number of surgeries before flowing to a minimal Clifford torus in infinite time. Furthermore, we show that we can construct a torus with any finite number of surgeries before convergence. Along the way, we prove many interesting subsidiary results and develop methods which should be useful in studying Lagrangian mean curvature f…
In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagrangian submanifold of N. Moreover there is an (n-1)-torus T^n-1 in T^n and a sequence of non-flat immersed minimal Lagrangian tori L_k in N, invariant under T^n-1 s.t. L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between…
It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in the complex projective plane may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in the complex projective plane, but the details require a careful treatment of both the "dressing construction" and the spectral data to deal with a loop of flat connexions which is quadratic in the loop paramet…
In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n\leq 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n\geq 3$. On the other hand, we prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any $n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are not Hamiltonian volume minimizing when $n\geq 3$ as well as the ca…
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. We use our examples of one-parameter families to show that the semi-flat metric on the mirror manifold proposed be N. Hitchin is not necessarily Ricci-flat in dimension …
Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebrated Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism. In the previous works the authors defined and studied a family of H-minimal Lagrangian submanifolds in complex space arising from intersections of Hermitian quadrics. Here we extend this cons…
Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost Kähler manifold. Special Lagrangians, and, more specificaly, fibrations by special Lagrangians play an important role in the context of the geometric mirror symmetry conjecture. However, these objects are rather scarce in nature. On the contrary, we show that HSLAG submanifolds, or fibrations, arise quite often. Many examples of HSLAG fibrations are provided by toric Käh-ler g…
We associate a periodic two-dimensional Schrodinger operator to every Lagrangian torus in CP^2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov-Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP^2 preserving the spectral curve. We also show that the highest flows on the space of smooth periodic solutions of the Tzizeica equation are given by the Novikov-Veselov hierarchy.
This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can use to "read off" a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings…
In this article I show that every special Lagrangian cone in C^3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU_3/SO_2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some examples found recently by Joyce are put into this context.
