arXiv research

In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…

The study classifies complete Lagrangian self-shrinkers in 4D space.

Proves rigidity of certain Lagrangian shrinkers using a pointwise approach.

The paper identifies special Lagrangian shapes in 4D space.

The paper proves Hessian estimates for specific geometric flows.

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Ca…

We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.

In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of $n$-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form sati…

Paper proves rigidity of certain 2D Lagrangian shapes in 4D space.

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangian…

Let $F_n :(Σ, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to …

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking Kähler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence…

In this paper, we generalize Colding and Minicozzi's work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover…

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…

The study classifies and proves rigidity of Legendrian self-shrinkers in 3D and 5D.

In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…

The paper constructs examples of Lagrangian flows using moment maps.

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…

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. I…

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…

Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow

The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.

Classifies regularity for Lagrangian mean curvature type equations.

In this paper, we first introduce the concept of $ξ$-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $λ$-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian $ξ$-submanifold in the complex $2$-p…

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a boun…

Compact shrinkers with curvature pinching conditions proven.

Proves existence of shrinkers via mean curvature flow.

Paper finds only one unique regular shrinker with 2 closed regions.

The study counts ends on shrinkers using geometric covering methods.

Study on stability of network flow shrinkers with findings on instability of specific shapes.

Researchers set entropy limits for specific types of self-shrinkers.

New theorem shows noncompact self shrinkers are unknotted.

Strong Frankel theorem for shrinkers in all dimensions.

Study bounds on self-shrinkers with bounded HA for applications.

Develops structure theory for Ricci shrinkers without curvature restrictions.

The paper proves bounded curvature for Kähler Ricci shrinker surfaces.

Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.

Survey of self-shrinkers with symmetry and new existence problems.

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

New theorems on compactness and finiteness for specific types of self-shrinkers.

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

Generalizes halfspace theorems to higher dimensions for self-shrinkers.

Compact self shrinkers in 3D are topologically standard.

Eigenvalue estimate for shrinkers in mean curvature flow.

We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form $\mathbb S^k\times\R^{n-k}\subset \R^{n+1}$. We use a connection between the stability operator and the quantum harmonic oscillator Ham…

Estimates the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.