# On energy gap phenomena of the Whitney sphere and related problems

@article{Luo2020OnEG, title={On energy gap phenomena of the Whitney sphere and related problems}, author={Yong Luo and Jiabin Yin}, journal={arXiv: Differential Geometry}, year={2020} }

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.

## 2 Citations

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

- Mathematics
- 2021

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

- Materials ScienceGeometriae Dedicata
- 2021

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

## References

SHOWING 1-10 OF 31 REFERENCES

### An energy gap phenomenon for the Whitney sphere

- MathematicsMathematische 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

- MathematicsAnnales 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

- Mathematics
- 1998

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

- Mathematics
- 1995

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

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1996

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

- Mathematics
- 2001

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

- Mathematics
- 1982

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

- Mathematics
- 2015

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

- Mathematics
- 1995

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

- Mathematics
- 2010

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…