Bila Nicoleta
One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.
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Using the symmetry group theory of second order PDEs, one finds the symmetry group associated to Tzitzeica surfaces partial differential equation. One studies the inverse problem and one shows that the Tzitzeica surfaces PDE is an Euler-Lagrange equation. One determines the variational symmetry group of the associated functional and one obtains the conservation laws of the Tzitzeica surfaces PDE. All these results shows that the Tzitzeica surfaces theory is strongly related to variational problems and hence it is a subject of global differential geometry.
A discrete analog of the Tzitzeica equation is found in the form of quad-equation. Its continuous symmetry is an inhomogeneous Narita--Bogoyavlensky type lattice equation which defines a discretization of the Sawada--Kotera equation. The integrability of these discretizations is proven by construction of the Lax representations.
We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus rec…
Given the Lagrangian fibration $T^4\to T^2$ and a Lagrangian submanifold, exhibiting an elliptic umbilic and supporting a flat line bundle, we study, in the context of mirror symmetry, the ``quantum'' corrections necessary to solve the monodromy of the holomorphic structure of the mirror bundle on the dual fibration.
We discuss the characterization of relative equilibria of Lagrangian systems with symmetry.
In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to an $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by Marianty Ionel. However, the classification of special Lagrangian submanifolds in $\mathbb{C}^4$ having this $SO(2)\rtimes S_3$ symmetry in that paper is incomplete. In the present paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with $SO(2)\rtimes S_3$-sy…
In this paper we summarize our recent work in the construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties and the symplectic Strominger-Yau-Zaslow conjecture, together with some new development. It is submittded to the Proceedings of the Conference in Symplectic Geometry and Mirror Symmetry held in KIAS, Seoul, Korea. The paper was finished in Jan. 2001, with revisions and added reference.
In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infinity algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered A infinity structure associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudo-isotopy of cyc…
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lagrangian 3-folds in C^3. This paper describes a construction of special Lagrangian m-folds in C^m which are fibred by (m-1)-submanifolds which are quadrics in Lagrangian planes R^m in C^m. Generically they have only discrete symmetry groups. Some of our examples have been previous…
This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian subm…
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).
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discre…
Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry
本文研究具有圆对称性的hyperkähler 4-流形中的拉格朗日平均曲率流，证明其可穿过有限时间奇点延续并收敛至special Lagrangians链，验证了Joyce猜想并指出奇点为颈缩。
In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introduce a notion of symmetry groups for objects of LP_d and introduce a reduction procedure that is closed in the category LP_d. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residua…
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both the Euler-Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper we demonstrate that the knowledge of this structure considerably eases finding the extremal curves for variational problems inva…
In this paper we will present Lagrangian and Hamiltonian $k$-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for $k$-symplectic Hamiltonian systems and $k$-symplectic Lagrangian systems.
In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincaré systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of $LDP_d$ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange--Poincaré systems as well as the one defined for non…
We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,α}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on col…
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