• Corpus ID: 219687442

On energy gap phenomena of the Whitney sphere and related problems

  title={On energy gap phenomena of the Whitney sphere and related problems},
  author={Yong Luo and Jiabin Yin},
  journal={arXiv: Differential Geometry},
In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c\geq0)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. We obtain some integral inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover we study Lagrangian surfaces in $\mathbb{C}^2$ satisfying $\nabla^*\nabla^*T=0$ and introduce a flow method related to them. 

On energy gap phenomena of the Whitney spheres in $\mathbb{C}^n$ or $\mathbb{CP}^n$

In [29] [19] Zhang, Luo and Yin initiated the study of Lagrangian submanifolds satisfying ∇T = 0 or ∇∇T = 0 in C or CP, where T = ∇∗h̃ and h̃ is the Lagrangian trace-free second fundamental form.

Gap theorems for Lagrangian submanifolds in complex space forms

In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying ∇∗T≡0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}



An energy gap phenomenon for the Whitney sphere

  • L. Zhang
  • Mathematics
    Mathematische Zeitschrift
  • 2020
In this paper, we study Lagrangian surfaces satisfying $$\nabla ^*T=0$$ ∇ ∗ T = 0 , where $$T=-2\nabla ^*(\check{A}\lrcorner \omega )$$ T = - 2 ∇ ∗ ( A ˇ ⌟ ω ) and $$\check{A}$$ A ˇ is the Lagrangian

Rigidity of closed CSL submanifolds in the unit sphere

  • Yong LuoLinlin Sun
  • Mathematics
    Annales de l'Institut Henri Poincaré C, Analyse non linéaire
  • 2022
A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in

Lagrangian submanifolds of $C^{n}$ with conformal Maslov form and the Whitney sphere

The Lagrangian submanifolds of the complex Euclidean space with conformal Maslov form can be considered as the Lagrangian version of the hypersurfaces of the Euclidean space with constant mean

The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers

In this paper we prove an existence and regularity theorem for la- grangian tori minimizing the Willmore functional in Euclidean four-space, R4, with the standard metric and symplectic structure.

Lagrangian submanifolds satisfying a basic equality

Abstract In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following

Closed conformal vector fields and Lagrangian submanifolds in complex space forms

We study a wide family of Lagrangian submanifolds in non flat complex space forms that we will call pseudoumbilical because of their geometric properties. They are determined by admitting a closed

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M

On geometrically constrained variational problems of the Willmore functional I: The Lagrangian-Willmore problem

In this paper, we study a kind of geometrically constrained variational problem of the Willmore functional. A surface l : Σ → C is called a Lagrangian–Willmore surface (in short, a LW surface) or a

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ℂℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over

Rigidity of entire self-shrinking solutions to curvature flows

Abstract We show (a) that any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℂn with the Euclidean metric is flat; (b) that any space-like entire graphic