arXiv research

Develops analytic tools for DSL, a geodesic equation in positive graph Lagrangians.

Study on evolving graphs of functions under mean curvature flow in R^n.

Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

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…

Geodesic connects Lagrangian graphs over torus in complex space.

Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.

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

The paper proves almost positive links can be filled with Lagrangian structures.

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

The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric $Υ$, which gives rise to a notion of geodesics. We study geodesics of positive $O_n(\mathbb{R})$ invariant…

Constructs geodesics near intersection points of Lagrangian submanifolds.

Analyzes geodesics in positive Lagrangian spaces on Riemannian manifolds.

We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.

In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…

Study finds many Lagrangian fillings for certain Legendrian links.

Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional…

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Characterizes Legendrian knots with exact Lagrangian fillings.

Positive braids have endless filling possibilities.

Study on radial solutions in higher dimensions, finding special cases.

Study of Legendrian links using Floer theory and cluster varieties.

We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic.

The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.

Continues study on special Lagrangian graphs and flow solutions.

Minimal Lagrangians in certain curved spaces are stable under specific flows.

A new Lagrangian method for graph neural networks accelerates state computation.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

Let Ωand \tildeΩ be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: Ω\to \tildeΩ such that the graph Σ= \{(x,f(x)): x \in Ω\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.

Lagrangian spheres develop singularities under flow, matching Whitney spheres.

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…

We prove the existence of Lagrangian fillings for $D_n$-type Legendrian links.

The paper connects Legendrian links to cluster algebras via microlocal methods.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.

We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.

Study Legendrian surfaces using N-graphs and flag moduli.

Develops a method to construct entire minimal graphs of odd dimensions.

Special Lagrangian submanifolds emerge from K3 surface collapse.

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.

In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.

Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.

The study explores surgeries on Lagrangian skeleta in 4-manifolds, connecting geometric and algebraic structures.