arXiv research

Ancient solutions and translators identified for Lagrangian flow.

Ancient Lagrangian flows get limited convex solutions.

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

The paper proves constant rank theorems for special Lagrangian equations.

Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.

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

In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

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.

Classifies regularity for Lagrangian mean curvature type equations.

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded…

We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

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

We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g>0, we can construct a real 6 dimensional Calabi-Yau cone M_g and a 3 dimensional special Lagrangian submanifold L^1_g in M_g which is…

The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in geometric analysis to construct new solutions to a given nonlinear partial diffe…

Maps in Carnot groups are equivalent to solutions of a PDE system.

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

Study on critical Lagrangian phase singularities in mean curvature flow.

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

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in…

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee,…

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.

The paper studies bifurcations in Lagrangian systems and geodesics.

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.

In this paper, we discuss the Lagrangian angles of a family of Lagrangian fibrations moved under mean curvature flow. In the case $n=1$, the angle function is shown to satisfy a degenerated partial differential equation. We prove that any smooth solution to the equation also corresponds to smooth foliation of curves un…

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 La…

This note surveys and compares results on the separation of variables construction for soliton solutions of curvature equations including the Kähler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.

We obtain some equations for Hamiltonian-minimal Lagrangian surfaces in CP^2 and give their particular solutions in the case of tori.

Proves existence of Lagrangian mean curvature flow solutions.

New proof for convex solutions of Monge-Ampère equation.

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

We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.

Develops methods to solve complex and real Hessian equations.

A new method solves complex constrained minimax problems.

We construct new examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow by extending the method of Joyce, Lee and Tsui. Those examples include examples in which the Lagrangian angle is arbitrarily small as the examples of Joyce, Lee and Tsui.

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we c…

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $γ$ in a differentiable manifold $M$ that are everywhere tangent to a smooth distribution $\mathcal D$ on $M$; such curves are called horizontal. We study the manifold structure of the…

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.

We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.

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.

LDDNN learns physical dynamics from data without exact solutions.

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the H…

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…