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Lagrangian immersions of homogeneous 3-manifolds in complex space forms
Last modified: 2014-03-13
Abstract
This is a report about joint work with X. Wang (Nankai University). We discuss Lagrangian submanifolds of the $3$-dimensional complex space forms. In the last decades many results about Lagrangian submanifolds have been obtained characterising several classes of submanifolds. Some are valid in all dimensions, while other are only valid in dimension $2$ or $3$. Nevertheless, even in low dimensions, many open questions remain.
On the other hand, also in the last decades, many people have started to study the geometry of some special $3$-dimensional homogeneous spaces. In particular many results have been obtained about how to study surfaces in such manifolds. Such manifolds can be divided into several classes depending of the dimension of the isometry group. They include \begin{enumerate}
\item the spaces with $6$-dimensional isometry group (Euclidean space, standard sphere $S^3$, hyperbolic space)
\item the spaces with $4$-dimensional isometry group (which consists of $\mathbb{S}^2 \times \mathbb R$, $\mathbb{H}^2 \times \mathbb R$, the Berger spheres, the Heisenberg group $\text{Nil}_3$ and the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$
\item the spaces with $3$-dimensional isometry group (which include amongst others the Lie group $Sol_3$)
\end{enumerate}
Of course equally interesting questions are whether any of these spaces can be isometrically immersed (locally or globally) as a Lagrangian submanifold in a complex projective space form, and whether such immersions, if they exist are necessarily unique.
A first result which can be formulated within this framework is the well known result by Ejiri.
\begin{thm}\label{thm1.1} Let $\phi$ be a minimal Lagrangian immersion from a 3-dimensional manifold with constant sectional curvature in the $3$-dimensional complex space form. Then either
\begin{itemize}
\item $M$ is totally geodesic, or
\item $M$ is congruent to a flat torus in the complex projective space.
\end{itemize}
\end{thm}
Note that without the assumption of minimality, the answer is not known. Even assuming that the immersion is a global one. There exist only some partial results, assuming additional assumptions.
If one looks at the other Thurston geometries (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$, the Lie group $\text{Sol}_3$, the product spaces $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$) the situatuion turns actually out to be more easy. Note that all these spaces are all quasi-Einstein, i.e. the Ricci tensor has an eigenvalue of multiplicity $2$ at every point and have constant scalar curvature.
In contrast to immersions of real space forms, a Lagrangian isometric immersion (even locally) of one of the following homogeneous 3-manifolds (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$ and the Lie group $\text{Sol}_3$) in a complex space form is always minimal. Moreover the only possibility is a unique isometric Lagrangian immersion of a Berger sphere in $\mathbb{CP}^3$.
\begin{thm}\label{thm1.2}
Let $\phi$ be a Lagrangian isometric immersion from (an open part of) one of the homogeneous 3-manifolds $M$ (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$ and the Lie group $\text{Sol}_3$) to a complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$. Then $c=1$, $M$ is a Berger sphere $\Sp^3_b(4/3, 1)$, $\phi$ is minimal. And up to an isometry of $\mathbb{CP}^3$, the immersion $\phi$ is unique.
\end{thm}
For the other two types of homogeneous 3-manifolds (the product spaces $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$), in order to be able to reach a conclusion, as in the constant sectional curvature case, it is necassary to assume that the immersion is minimal. In that case, we have
\begin{thm}\label{thm1.3}
Let $\phi$ be a minimal Lagrangian isometric immersion from an (open part of) $\mathbb{S}^2\times \mathbb{R}$ to a 3-dimensional complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$. Then $c=1$, $\phi$ is obtained as the Calabi product of the totally geodesic minimal immersion of $\mathbb{S}^2$ into $\mathbb CP^2$ and a point.
\end{thm}
\begin{thm}\label{thm1.4}
There does not exist a minimal Lagrangian isometric immersion from an open part of $\mathbb{H}^2\time \mathbb{R}$ into the 3-dimensional complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$.
\end{thm}
On the other hand, also in the last decades, many people have started to study the geometry of some special $3$-dimensional homogeneous spaces. In particular many results have been obtained about how to study surfaces in such manifolds. Such manifolds can be divided into several classes depending of the dimension of the isometry group. They include \begin{enumerate}
\item the spaces with $6$-dimensional isometry group (Euclidean space, standard sphere $S^3$, hyperbolic space)
\item the spaces with $4$-dimensional isometry group (which consists of $\mathbb{S}^2 \times \mathbb R$, $\mathbb{H}^2 \times \mathbb R$, the Berger spheres, the Heisenberg group $\text{Nil}_3$ and the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$
\item the spaces with $3$-dimensional isometry group (which include amongst others the Lie group $Sol_3$)
\end{enumerate}
Of course equally interesting questions are whether any of these spaces can be isometrically immersed (locally or globally) as a Lagrangian submanifold in a complex projective space form, and whether such immersions, if they exist are necessarily unique.
A first result which can be formulated within this framework is the well known result by Ejiri.
\begin{thm}\label{thm1.1} Let $\phi$ be a minimal Lagrangian immersion from a 3-dimensional manifold with constant sectional curvature in the $3$-dimensional complex space form. Then either
\begin{itemize}
\item $M$ is totally geodesic, or
\item $M$ is congruent to a flat torus in the complex projective space.
\end{itemize}
\end{thm}
Note that without the assumption of minimality, the answer is not known. Even assuming that the immersion is a global one. There exist only some partial results, assuming additional assumptions.
If one looks at the other Thurston geometries (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$, the Lie group $\text{Sol}_3$, the product spaces $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$) the situatuion turns actually out to be more easy. Note that all these spaces are all quasi-Einstein, i.e. the Ricci tensor has an eigenvalue of multiplicity $2$ at every point and have constant scalar curvature.
In contrast to immersions of real space forms, a Lagrangian isometric immersion (even locally) of one of the following homogeneous 3-manifolds (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$ and the Lie group $\text{Sol}_3$) in a complex space form is always minimal. Moreover the only possibility is a unique isometric Lagrangian immersion of a Berger sphere in $\mathbb{CP}^3$.
\begin{thm}\label{thm1.2}
Let $\phi$ be a Lagrangian isometric immersion from (an open part of) one of the homogeneous 3-manifolds $M$ (the Berger spheres, the Heisenberg group $\text{Nil}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb R)$ and the Lie group $\text{Sol}_3$) to a complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$. Then $c=1$, $M$ is a Berger sphere $\Sp^3_b(4/3, 1)$, $\phi$ is minimal. And up to an isometry of $\mathbb{CP}^3$, the immersion $\phi$ is unique.
\end{thm}
For the other two types of homogeneous 3-manifolds (the product spaces $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$), in order to be able to reach a conclusion, as in the constant sectional curvature case, it is necassary to assume that the immersion is minimal. In that case, we have
\begin{thm}\label{thm1.3}
Let $\phi$ be a minimal Lagrangian isometric immersion from an (open part of) $\mathbb{S}^2\times \mathbb{R}$ to a 3-dimensional complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$. Then $c=1$, $\phi$ is obtained as the Calabi product of the totally geodesic minimal immersion of $\mathbb{S}^2$ into $\mathbb CP^2$ and a point.
\end{thm}
\begin{thm}\label{thm1.4}
There does not exist a minimal Lagrangian isometric immersion from an open part of $\mathbb{H}^2\time \mathbb{R}$ into the 3-dimensional complex space form $\bar{M}^3(4c), c\in\{-1,0,1\}$.
\end{thm}