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Two-dimensional Lagrangian singularities and bifurcations of gradient lines I  [PDF]
G. Marelli
Mathematics , 2007,
Abstract: Motivated by mirror symmetry, we consider a Lagrangian fibration $X\to B$ and Lagrangian maps $f:L\hookrightarrow X\to B$, when $L$ has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood of the unstable singularity, when slightly perturbed. The integral curves of $\nabla f_x$, for $x\in B$, where $f_x(y)=f(y)-x\cdot y$, called ``gradient lines'', are then introduced, and a study of them, in order to analyse their bifurcation locus, is carried out.
Surgery on Lagrangian and Legendrian Singularities  [PDF]
Mikhail Entov
Mathematics , 1997,
Abstract: Let $\pi : E\to M$ be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian submanifold of $E$. In the paper we address the following question: how can one simplify the singularities of the projection $\pi: L \to M$ by a Hamiltonian isotopy of $L$ inside $E$? We give an answer in the case when $dim L = 2$ and both $L$ and $M$ are orientable. A weaker version of the result is proved in the higher-dimensional case. Similar results hold in the contact category. As a corollary one gets an answer to one of the questions of V.Arnold about the four cusps on the caustic in the case of the Lagrangian collapse. As another corollary we disprove Y.Chekanov's conjecture about singularities of the Lagrangian projection of certain Lagrangian tori in $R^4$.
Singularities and bifurcations of 3-dimensional Poisson structure  [PDF]
J. -P. Dufour,M. Zhitomirskii
Mathematics , 1998,
Abstract: We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable 1-forms on $R^3
Rigid and Complete Intersection Lagrangian Singularities  [PDF]
Christian Sevenheck,Duco van Straten
Mathematics , 2003,
Abstract: In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open swallowtails of dimension greater or equal than two. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersection in dimension greater than two cannot be regular in codimension one.
Singularities of special Lagrangian submanifolds  [PDF]
Dominic Joyce
Mathematics , 2003,
Abstract: We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities. That is, near each singular point x, X is modelled on an SL cone C in C^m with isolated singularity at 0. We also discuss directions for future research, and give a list of open problems.
Singularities of the Lagrangian Mean Curvature Flow  [PDF]
Andrew A. Cooper
Mathematics , 2015,
Abstract: In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asymptotic to special Lagrangian cones; hence all type II singularities are modeled by unions of special Lagrangian cones.
Classification of Singularities and Bifurcations of Critical Points of Even Functions  [PDF]
E. A. Kudryavtseva,E. Lakshtanov
Mathematics , 2012,
Abstract: Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and conditions for versal deformations are given in terms of partial derivatives (not requiring a preliminary reduction to a canonical form).
Recent Progress on Singularities of Lagrangian Mean Curvature Flow  [PDF]
André Neves
Mathematics , 2010,
Abstract: We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.
Lagrangian singularities of steady two-dimensional flow  [PDF]
W. Pauls,T. Matsumoto
Physics , 2004, DOI: 10.1080/03091920412331312411
Abstract: The Lagrangian complex-space singularities of the steady Eulerian flow with stream function $\sin x_1 \cos x_2$ are studied by numerical and analytical methods. The Lagrangian singular manifold is analytic. Its minimum distance from the real domain decreases logarithmically at short times and exponentially at large times.
Singularities of Lagrangian mean curvature flow: monotone case  [PDF]
Andre' Neves
Mathematics , 2006,
Abstract: We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this result by showing that connected components of the rescaled flow converge to an area-minimizing cone, as opposed to possible non-area minimizing union of Slag cones. In the last section, we give specific examples for which such singularity formation occurs.
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