arXiv research

New tan-concavity property for Lagrangian phase operators helps in studying dHYM metrics.

Paper proves estimates for Lagrangian flow singularities.

Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.

Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.

Study on critical Lagrangian phase singularities in mean curvature flow.

Study shows how neck pinches occur in Lagrangian flows and their continuation.

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature f…

Introduces a new phase space for 2D supersymmetric sigma models.

On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. For three-dimensional phase space the concept of vector hamiltonian and vector lagrangian is entered.

The paper proves Hessian estimates for specific geometric flows.

We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian …

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Constructing translating solitons from Lagrangian Grim Reapers.

We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…

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…

Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.

Classifies regularity for Lagrangian mean curvature type equations.

In this paper, we firstly prove that every hyper-Lagrangian submanifold $L^{2n} (n > 1)$ in a hyperkähler $4n$-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in $\mathbb{R}^4$. Last but not least, …

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagran…

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition …

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a larg…

Global minimizers exist for Tonelli Lagrangians on half-Lie groups.

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

Paper proves gradient estimates for Lagrangian mean curvature equation.

Paper develops estimates for Lagrangian phase changes in 2D.

Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.

Estimates for special Lagrangian curvature equations in critical and convex cases.

Derives Hessian estimates for Lagrangian mean curvature equation.

New special Lagrangian submanifolds with cylindrical tangent cones are constructed.

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant m…

Convex solutions to a specific equation are smooth when the phase is smooth enough.

The Samuelson condition is not satisfied by tangent lines of quadratic curves.

Paper doubles Hessian estimates for special Lagrangian equation with constraints.

Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.

Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.

We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

Paper solves flat bi-Lagrangian structure problems in ray space.

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

Study on deformations of special Lagrangians with boundary in Calabi-Yau manifolds.

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equ…

Around mid-1970s W. M. Tulczyjew discovered an approach which brings the two formalisms under a common geometric roof: the dynamics of a particle with configuration space $X$ is determined by a Lagrangian submanifold $D$ of $TT^*X$ (the total tangent space of $T^*X$), and the description of $D$ by its Hamiltonian $H$: …

Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.

The paper studies singularities in a complex flow related to mean curvature.

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.