Hamiltonian Stationary Lagrangian Surfaces with Non-Negative Gaussian Curvature in Kähler-Einstein Surfaces

Patrik Coulibaly
e-mail: [email protected]

Abstract

In this paper, we give some simple conditions under which a Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold must have a Euclidean factor or be a fiber bundle over a circle. We also characterize the Hamiltonian stationary Lagrangian surfaces whose Gaussian curvature is non-negative and whose mean curvature vector is in some $L^{p}$ space when the ambient space is a simply connected complex space form.

1 Introduction

Let $(M,g,J)$ be a Kähler manifold of complex dimension $n$ . $M$ carries a natural symplectic structure given by the closed $2$ -form $\omega$ which is defined by $\omega(X,Y)=g(JX,Y)$ for $X,Y\in T_{p}M$ . We say that a Lagrangian submanifold $L\subset M$ is Hamiltonian stationary if it is a critical point of the volume functional under compactly supported Hamiltonian deformations, i.e. variations for which the variational vector field is of the form $V=J\nabla f$ for some $f\in C^{\infty}_{c}(L)$ . In [14], Oh calculated the Euler-Lagrange equation of the variational problem and found that Hamiltonian stationary Lagrangian submanifolds are characterised by

\delta\alpha_{H}=0,

or equivalently by

\text{div}_{L}(JH)=0,

where $H$ denotes the mean curvature vector of $L$ , which we define as the trace of its second fundamental form $A$ , i.e. $H\mathrel{\mathop{\mathchar 58\relax}}=Tr_{g}A$ , $\alpha_{H}$ is the differential $1$ -form on $L$ defined by $\alpha_{H}\mathrel{\mathop{\mathchar 58\relax}}=\iota_{H}\omega=g(JH,\cdot)$ and $\delta$ is the co-differential operator on $L$ induced by the metric $g$ .

By a theorem of Dazord (see, for example, Theorem 2.1 in [14]), in any Kähler manifold $M$ , the restriction of the Ricci form $ric_{M}$ of $M$ to $L$ is given by $d\alpha_{H}$ . When $M$ is Kähler-Einstein, i.e. $ric_{M}=c\omega$ for some constant $c$ , then the differential $1$ -form $\alpha_{H}$ is closed and thus defines a cohomology class in H ${}_{dR}^{1}(L)\cong H^{1}(L;\mathbb{R})$ on any Lagrangian submanifold. Therefore, $\alpha_{H}$ is both closed and co-closed, hence harmonic, on any Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold.

In [2], Arsie proved that when $M$ is a Calabi-Yau manifold, then $\frac{1}{\pi}\alpha_{H}$ represents an integral cohomology class of $L$ called the Maslov class. Therefore, we will refer to $\mu=\frac{1}{\pi}\alpha_{H}$ as the Maslov form of $L$ .

Let $NL$ denote the normal bundle of $L$ in $M$ and $\Gamma(NL)$ denote the collection of smooth sections of $NL$ . Also, for any point $x\in L$ and vector $\overline{X}\in T_{x}M$ , let $\overline{X}^{\perp}$ denote the projection of $\overline{X}$ onto $N_{x}L$ and let $\overline{\nabla}$ denote the Levi-Civita connection on $M$ . Then, there is a connection $\nabla^{\perp}$ in $NL$ that is given by

\nabla^{\perp}_{X}V\mathrel{\mathop{\mathchar 58\relax}}=\left(\overline{\nabla}_{X}V\right)^{\perp}

for any normal vector field $V\in\Gamma(NL)$ and tangent vector $X\in T_{x}L$ . We say that a normal vector field $V\in\Gamma(NL)$ is parallel if $\nabla^{\perp}V\equiv 0$ .

For any point $x\in L$ and vector $\overline{X}\in T_{x}M$ , let $\overline{X}^{\top}$ denote the projection of $\overline{X}$ onto $T_{x}L$ . Then, since $L$ is Lagrangian and $\nabla J=0$ , we have that

\displaystyle\nabla_{X}JV=\left(\overline{\nabla}_{X}JV\right)^{\top}=\left(J\overline{\nabla}_{X}V\right)^{\top}=J\left(\overline{\nabla}_{X}V\right)^{\perp}=J\nabla^{\perp}_{X}V,

for any normal vector field $V\in\Gamma(NL)$ and tangent vector $X\in T_{x}L$ . Therefore, $JV$ is parallel if and only if $V$ is parallel.

We say that $L$ has parallel second fundamental form if

\displaystyle(\nabla_{X}A)(Y,Z)=\nabla^{\perp}_{X}A(Y,Z)-A(\nabla_{X}Y,Z)-A(Y,\nabla_{X}Z)

vanishes for all $X,Y,Z\in\Gamma(TL)$ .

We present our results in two separate sections. In Section 2, we consider a complete, connected Hamiltonian stationary Lagrangian submanifold $L$ of arbitrary dimension $n$ inside a Kähler-Einstein manifold. We introduce a set of conditions, most of which consist of the non-negativity of the Ricci curvature of $L$ in the direction of $JH$ and some pointwise or integral control over the absolute value of $H$ , that allows us to combine the Bochner formula for the harmonic $1$ -form $\alpha_{H}$ and some Liouville-type theorems to deduce that $H$ must be parallel in the normal bundle of $L$ . The existence of a non-trivial global parallel vector field can restrict both the topology and the geometry of a manifold significantly. For example, if $L$ is simply connected, then it must be isometric to a Riemannian product of the form $N\times\mathbb{R}$ . As for a purely topological consequence, if $L$ is not diffeomorphic to such a product, then it must admit a circle action whose orbits are not homologous to zero. In Section 3, we restrict our attention to the case when $n=2$ . We also strengthen our assumptions by requiring that our surface has non-negative Gaussian curvature which allows us to describe explicitly all complete, connected Hamiltonian stationary Lagrangian surfaces in $\mathbb{C}^{2}$ , $\mathbb{C}\mathbb{P}^{2}$ and in $\mathbb{C}\mathbb{H}^{2}$ that has non-negative Gaussian curvature and whose mean curvature vector is in some $L^{p}$ space.

The author would like to express his gratitude to Prof. Jingyi Chen for his invaluable suggestions and support, which were essential to the completion of this paper.

2 Hamiltonian Stationary Lagrangians in Kähler-Einstein Manifolds and the Bochner Method

Let $\mathcal{A}$ denote any of the following sets of assumptions:

1.

$Ric_{L}(JH,JH)\geq 0$ and $\mathinner{\!\left\lvert H\right\rvert}\in L^{p}$ for some $p\in(2,\infty)$ ;
2.

$L$ has non-negative Ricci curvature and $\mathinner{\!\left\lvert H\right\rvert}\in L^{p}$ for some $p\in(0,\infty)$ ;
3.

$L$ is oriented, $Ric_{L}(JH,JH)\geq 0$ and $\mathinner{\!\left\lvert H\right\rvert}\rightarrow c\mathrel{\mathop{\mathchar 58\relax}}=\inf_{L}\mathinner{\!\left\lvert H\right\rvert}$ as $r(x)\rightarrow\infty$ where $r(x)\mathrel{\mathop{\mathchar 58\relax}}=d(x_{0},x)$ is the distance function on $L$ relative to a fixed point $x_{0}\in L$ ;
4.

$Ric_{L}(JH,JH)\geq 0$ and there exists a point $x_{0}\in L$ , a non-decreasing function $f\mathrel{\mathop{\mathchar 58\relax}}[0,\infty)\mapsto[0,\infty)$ , constants $C,R>0$ and $p\in(2,\infty)$ such that $\mathinner{\!\left\lvert H(x)\right\rvert}\leq f(r(x))$ for all $x\in L$ and

$\displaystyle\frac{f(r)^{p}Vol(B_{r}(x_{0}))}{r^{2}\log(r)}\leq C$ (1)

whenever $r\geq R$ . Here, $B_{r}(x_{0})$ denotes the geodesic ball in $L$ of radius $r$ around the point $x_{0}\in L$ ;
5.

$L$ has conformal Maslov form, i.e. the vector field $JH$ is conformal.

If at least one of the sets of assumptions labelled (1)–(5) is satisfied, we say that $\mathcal{A}$ is satisfied.

We can state the main result of this section as follows.

Theorem 2.1.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. If $L$ satisfies $\mathcal{A}$ , then

(a)

$H$ is parallel and thus has constant length;
(b)

$Ric_{L}(JH,\cdot)$ vanishes identically, so if there exists a point $x\in L$ such that $Ric_{L}|_{x}$ is non-degenerate, then $L$ must be minimal;
(c)

and the scalar curvature of $L$ must be constant along the integral curves of $JH$ .

The growth bound (1) from condition (4) is satisfied, for example, when $L$ has quadratic volume growth and $\mathinner{\!\left\lvert H\right\rvert}$ does not grow faster than $\log(r)^{\frac{1}{2p}}$ at infinity for some $1<p<\infty$ . In particular, it is satisfied when $L$ has quadratic volume growth and $\mathinner{\!\left\lvert H\right\rvert}\in L^{\infty}$ . Therefore, we have the following corollary.

Corollary 2.2.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. If $Ric(JH,JH)\geq 0$ , $\mathinner{\!\left\lvert H\right\rvert}\in L^{\infty}$ and $L$ has quadratic volume growth, then the conclusions of Theorem 2.1 hold; in particular, $\mathinner{\!\left\lvert H\right\rvert}$ must be constant.

Remark.

This phenomenon is related to the notion of parabolicity of a manifold. We say that a manifold is (strongly) parabolic if it does not admit a negative, non-constant subharmonic function, i.e. if $f<0$ and $\Delta f\geq 0$ , then it must be constant. It is easy to see that a parabolic manifold does not admit a non-constant subharmonic function that is bounded from above. A sufficient condition¹¹1As it is discussed in [6] after Corollary 7.7, when the Ricci curvature is non-negative, then this condition is also necessary. for the parabolicity of a manifold was given by Karp in [9], which implies, for example, that every complete, non-compact manifold with quadratic volume growth is parabolic.

The main restriction imposed on $L$ by the conclusion of the Theorem 2.1 is that $JH$ is parallel since the existence of a non-trivial global parallel vector field restricts the topology of a manifold significantly. For example, the following result of Welsh [19], states that the existence of a complete non-trivial global parallel vector field forces the existence of a circle action whose orbits are not real homologous to zero. By a complete vector field, we mean a vector field whose integral curves are defined for all time.

Theorem 2.3 (Welsh [19]).

Suppose that $M$ is a Riemannian manifold that admits a non-zero complete parallel vector field. Then either $M$ is diffeomorphic to the product of a Euclidean space with some other manifold, or else there is a circle action on $M$ whose orbits are not real homologous to zero. Moreover, if $M$ is not diffeomorphic to the product of a Euclidean space with some other manifold and its first integral homology class is finitely generated, then $M$ is a fiber bundle over a circle with finite structure group.

Combining Theorem 2.1 and Theorem 2.3 gives us the following corollary.

Corollary 2.4.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. If $L$ is not minimal and it satisfies $\mathcal{A}$ , then $L$ is diffeomorphic to the product of a Euclidean space with some other manifold or there is a circle action on $M$ whose orbits are not real homologous to zero. Moreover, it satisfies the conclusion of Theorem 2.1; and if it is not diffeomorphic to the product of a Euclidean space with some other manifold and its first integral homology class is finitely generated, then $M$ is a fiber bundle over a circle with finite structure group.

Proof.

Suppose that $L$ is a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold that satisfies $\mathcal{A}$ . Then all the assumptions of Theorem 2.1 are satisfied thus all of its conclusions hold. In particular, $JH$ is parallel so its integral curves are geodesics. Since $L$ is complete, all of its geodesics are defined for all $t\in\mathbb{R}$ and we see that $JH$ is a complete vector field. Therefore, when $L$ is not minimal, $JH$ is a non-zero complete parallel vector field on $L$ and we can apply Theorem 2.3 to finish the proof. ∎

When $L$ is not minimal, we can use Corollary 2.4 to establish the existence of a circle action on $L$ whose orbits are not real homologous to zero but only if $L$ is not diffeomorphic to a product of $\mathbb{R}$ and some other manifold. It turns out that when $L$ is simply connected then the existence of such a splitting is guaranteed. Moreover, $L$ can be split in such a way isometrically.

Corollary 2.5.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. If $L$ is not minimal and it satisfies $\mathcal{A}$ , then its universal cover $\pi\mathrel{\mathop{\mathchar 58\relax}}\tilde{L}\to L$ equipped with the pull-back metric is isometric to $N\times\mathbb{R}$ for some totally geodesic submanifold $N$ of $\tilde{L}$ .

Proof.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold that satisfies $\mathcal{A}$ . Suppose that $L$ is not minimal. Then, by Theorem 2.1, $JH$ is a non-zero parallel vector field.

Let $\pi\mathrel{\mathop{\mathchar 58\relax}}\tilde{L}\to L$ denote the universal cover of $L$ which we equip with the pull-back metric. This makes $\pi$ into a Riemannian covering. We know that $d\alpha_{H}=\delta\alpha_{H}=0$ and since $\pi$ is a local isometry, we must also have $d\tilde{\alpha}_{H}=\delta\tilde{\alpha}_{H}=0$ for $\tilde{\alpha}_{H}=\pi^{*}\alpha_{H}$ . Define $\widetilde{JH}=(\tilde{\alpha}_{H})^{\sharp}$ . Then $JH=\pi_{*}\widetilde{JH}$ so $\widetilde{JH}$ must also be a parallel vector field. Since $\tilde{L}$ is simply connected and $\tilde{\alpha}_{H}$ is closed, there exists a smooth function $f\in C^{\infty}(\tilde{L})$ such that $\tilde{\alpha}_{H}=df$ or equivalently $\widetilde{JH}=\nabla f$ .

Since $\widetilde{JH}=\nabla f$ is parallel, by Lemma 2.3. in [16], $f$ is an affine function in the sense that $f\circ\gamma\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to\mathbb{R}$ satisfies

f\circ\gamma(\lambda t_{1}+(1-\lambda)t_{2})=\lambda f\circ\gamma(t_{1})+(1-\lambda)f\circ\gamma(t_{2})

for all maximal unit speed geodesics $\gamma$ in $\widetilde{L}$ , $\lambda\in(0,1)$ and $t_{1},t_{2}\in\mathbb{R}$ . Also, since $\widetilde{JH}=\nabla f$ is non-zero, $f$ is a non-trivial affine function so, by a theorem of Innami [8], $f^{-1}(0)$ is a totally geodesic submanifold of $\tilde{L}$ and $f^{-1}(0)\times\mathbb{R}$ is isometric to $\tilde{L}$ . ∎

In order to prove Theorem 2.1, we need the following lemma.

Lemma 2.6.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. If $L$ satisfies $\mathcal{A}$ , then $\mathinner{\!\left\lvert H\right\rvert}$ is constant.

Proof.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold. Then $\alpha_{H}=(JH)^{\flat}$ is closed so we can apply the Bochner formula [15, p. 207] for $\mathinner{\!\left\lvert\alpha_{H}\right\rvert}^{2}=\mathinner{\!\left\lvert JH\right\rvert}^{2}=\mathinner{\!\left\lvert H\right\rvert}^{2}$ to get

	$\displaystyle\frac{1}{2}\Delta_{L}\mathinner{\!\left\lvert H\right\rvert}^{2}$	$\displaystyle=\langle JH,\nabla\text{div}_{L}JH\rangle+Ric_{L}(JH,JH)+\mathinner{\!\left\lvert\nabla JH\right\rvert}^{2}$
		$\displaystyle=Ric_{L}(JH,JH)+\mathinner{\!\left\lvert\nabla^{\perp}H\right\rvert}^{2}$

First, we observe that if $\mathcal{A}$ is one of the conditions (1)–(4) , then $Ric_{L}(JH,JH)\geq 0$ and the function $\mathinner{\!\left\lvert H\right\rvert}^{2}$ is clearly subharmonic. Since compact manifolds do not admit non-constant subharmonic functions, we may assume without the loss of generality that $L$ is non-compact when $\mathcal{A}$ is one of the conditions (1)–(4).

If $\mathinner{\!\left\lvert H\right\rvert}^{2}$ is in $L^{p}$ for some $1<p<\infty$ , then by a well-known result of Yau [21], $\mathinner{\!\left\lvert H\right\rvert}^{2}$ is constant. Therefore, condition (1) implies that $\mathinner{\!\left\lvert H\right\rvert}$ is constant.

If $L$ has non-negative Ricci curvature, then by a result of Li and Schoen (Theorem 2.2. in [12]) it does not admit any non-negative $L^{p}$ subharmonic function for all $0<p<\infty$ . Thus, condition (2) implies that $\mathinner{\!\left\lvert H\right\rvert}$ is constant.

Now, assume that condition (3) is satisfied. In [1], Alías, Caminha and do Nascimento prove that every non-negative subharmonic function that converges to 0 at infinity on a connected, oriented, complete and non-compact Riemannian manifold must be identically zero. Applying this maximum principle to the function $f=\mathinner{\!\left\lvert H\right\rvert}^{2}-c$ gives us that $\mathinner{\!\left\lvert H\right\rvert}^{2}\equiv c$ . Therefore, we conclude that condition (3) also implies that $\mathinner{\!\left\lvert H\right\rvert}$ is constant.

Next, assume that condition (4) is satisfied. Since $f$ is a non-negative and non-decreasing function,

	$\displaystyle\frac{1}{r^{2}\log r}\int_{B_{r}(x_{0})}\mathinner{\!\left\lvert H\right\rvert}^{p}dV$	$\displaystyle\leq\frac{1}{r^{2}\log r}\int_{B_{r}(x_{0})}f(r)^{p}dV$
		$\displaystyle=\frac{f(r)^{p}Vol(B_{r}(x_{0}))}{r^{2}\log r}$
		$\displaystyle\leq C$

whenever $r\geq R$ . Therefore,

\limsup_{r\to\infty}{\frac{1}{r^{2}\log r}\int_{B_{r}(x_{0})}\left(\mathinner{\!\left\lvert H\right\rvert}^{2}\right)^{\frac{p}{2}}dV}<\infty

However, in [9], Karp showed that every non-negative non-constant subharmonic function $g$ on a complete non-compact Riemannian manifold satisfies

\limsup_{r\to\infty}{\frac{1}{r^{2}\log r}\int_{B_{r}(x)}g^{q}dV}=\infty

for all $q\in(1,\infty)$ and center $x$ . Therefore, $\mathinner{\!\left\lvert H\right\rvert}^{2}$ must be constant and we can conclude that condition (4) also implies that $\mathinner{\!\left\lvert H\right\rvert}$ is constant.

Finally, suppose that $JH$ is conformal. Then, since $JH$ is divergence-free,

\mathcal{L}_{JH}\hskip 2.0ptg=\frac{2}{n}\text{div}(JH)g=0.

Therefore, the vector field $JH$ is in fact Killing and the tensor $\langle\nabla JH,\boldsymbol{\cdot}\hskip 2.0pt\rangle$ is skew-symmetric. Since the dual $1$ -form $\alpha_{H}$ is closed, we also know that $\langle\nabla JH,\boldsymbol{\cdot}\hskip 2.0pt\rangle$ is symmetric, and hence it must be zero. Therefore, $JH$ is parallel which implies that it must also have constant length. We can conclude that if $L$ has conformal Maslov class, then $\mathinner{\!\left\lvert H\right\rvert}$ must be constant, which completes the proof. ∎

Now, we can prove Theorem 2.1.

Proof of Theorem 2.1.

Let $L$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein manifold that satisfies $\mathcal{A}$ . Then, as in the proof of Lemma 2.6, we have that

\displaystyle\frac{1}{2}\Delta_{L}\mathinner{\!\left\lvert H\right\rvert}^{2}=Ric_{L}(JH,JH)+\mathinner{\!\left\lvert\nabla^{\perp}H\right\rvert}^{2}.

(2)

By Lemma 2.6, $\mathinner{\!\left\lvert H\right\rvert}$ is constant so the left-hand side of equation (2) vanishes identically. When $\mathcal{A}$ is one of the conditions (1)–(4), then we have two non-negative terms on the right-hand side so they must each vanish identically, i.e. we must have that $Ric_{L}(JH,JH)\equiv 0$ and $\mathinner{\!\left\lvert\nabla^{\perp}H\right\rvert}^{2}\equiv 0$ . When $JH$ is conformal, then by the same argument that we used in the proof of Lemma 2.6, $JH$ is parallel. So $\mathinner{\!\left\lvert\nabla^{\perp}H\right\rvert}^{2}\equiv\mathinner{\!\left\lvert\nabla JH\right\rvert}^{2}\equiv 0$ which forces $Ric_{L}(JH,JH)\equiv 0$ . Therefore, we can conclude that $H$ is parallel and $Ric_{L}(JH,JH)$ is identically zero whenever $\mathcal{A}$ is satisfied. This proves $(a)$ .

Recalling the Weitzenböck formula [15, p. 211], we have

\displaystyle\Delta\alpha_{H}=-Tr_{g}(\nabla^{2}\alpha_{H})+Ric_{L}(JH,\cdot)

where $\Delta$ is the Hodge-Laplacian acting on differential $1$ -forms. Since $\alpha_{H}$ is both harmonic and parallel, we have that

\Delta\alpha_{H}=Tr_{g}(\nabla^{2}\alpha_{H})=0

and thus $Ric_{L}(JH,\cdot)$ must also vanish identically. Let us also assume that there exists a point $x\in L$ such that $Ric_{L}|_{x}$ is non-degenerate. Since $Ric_{L}|_{x}(JH|_{x},JH|_{x})=0$ , we must have that $JH|_{x}=0$ . However, we know that $JH$ has constant length so $JH$ must vanish identically and thus $L$ is minimal. This proves $(b)$ .

Let us also recall the contracted Bianchi identity (Proposition 7.18. [11])

\displaystyle\frac{1}{2}dS_{L}=Tr_{g}\nabla Ric_{L}

(3)

where $S_{L}$ is the scalar curvature of $L$ . The trace is taken on the first and the third indices, i.e. given a local orthonormal frame $E_{1},\dots,E_{n}$ , equation (3) reads as

\displaystyle\frac{1}{2}dS_{L}

\displaystyle=(\nabla_{E_{i}}Ric_{L})(\cdot,E_{i}).

Therefore, plugging $JH$ into equation (3) gives us that

	$\displaystyle\frac{1}{2}dS_{L}(JH)$	$\displaystyle=(\nabla_{E_{i}}Ric_{L})(JH,E_{i})$
		$\displaystyle=E_{i}Ric_{L}(JH,E_{i})-Ric_{L}(\nabla_{E_{i}}JH,E_{i})-Ric_{L}(JH,\nabla_{E_{i}}E_{i})$
		$\displaystyle=0.$

The first and the third terms vanish since $Ric_{L}(JH,\cdot)\equiv 0$ , while the second term is zero because $JH$ is parallel. So we conclude that the scalar curvature must be constant along the integral curves of $JH$ , which completes the proof of $(c)$ . ∎

3 Hamiltonian Stationary Lagrangian Surfaces with Non-Negative Gaussian Curvature in Kähler-Einstein surfaces

Let $\Sigma$ denote a complete connected Hamiltonian stationary Lagrangian surface isometrically immersed in a Kähler-Einstein surface $M$ .

In order to obtain a characterization that is more explicit than the one given by Theorem 2.1, we adjust our sets of assumptions from before. Let $\mathcal{A}^{\prime}$ denote any of the following sets of assumptions:

1.

$\Sigma$ has non-negative Gaussian curvature and $\mathinner{\!\left\lvert H\right\rvert}\in L^{p}$ for some $p\in(0,\infty)$ ;
2.

$\Sigma$ is oriented, it has non-negative Gaussian curvature and $\mathinner{\!\left\lvert H\right\rvert}\rightarrow c\mathrel{\mathop{\mathchar 58\relax}}=\inf_{\Sigma}\mathinner{\!\left\lvert H\right\rvert}^{2}$ as $r(x)\rightarrow\infty$ ;
3.

$\Sigma$ has non-negative Gaussian curvature and the growth condition (1) is satisfied;
4.

$\Sigma$ has non-negative Gaussian curvature and conformal Maslov form.

If at least one of the sets of assumptions labelled (1)–(4) is satisfied, we say that $\mathcal{A}^{\prime}$ is satisfied.

We will treat the cases when $\Sigma$ is compact and when it is non-compact separately.

Theorem 3.1.

Let $M$ be a Kähler-Einstein surface and let $\Sigma$ be a closed connected Hamiltonian stationary Lagrangian surface in $M$ that satisfies $\mathcal{A}^{\prime}$ . If $\Sigma$ is orientable, then it is

•

a flat torus or
•

a minimal sphere.

If $\Sigma$ is not orientable, then it is

•

a flat Klein bottle or
•

a minimal projective plane.

In both cases, $\Sigma$ has parallel mean curvature.

Theorem 3.2.

Let $M$ be a Kähler-Einstein surface. If $\Sigma$ is a complete, connected non-compact Hamiltonian stationary Lagrangian surface in $M$ satisfying $\mathcal{A}^{\prime}$ , then it has parallel mean curvature and it is

•

isometric to $\mathbb{R}^{2}$ ,
•

diffeomorphic to $\mathbb{R}^{2}$ and is minimal or
•

it is flat and its fundamental group is isomorphic to $\mathbb{Z}$ .

We start by proving the following lemma.

Lemma 3.3.

Let $\Sigma$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein surface $M$ . If $\Sigma$ satisfies $\mathcal{A}^{\prime}$ then it has parallel mean curvature and it is also flat or minimal.

Proof.

Let $\Sigma$ be a complete, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein surface $M$ . Since dim ${}_{\mathbb{R}}(\Sigma)=2$ , its curvature is entirely determined by its Gaussian curvature $K$ and, in particular,

Ric_{\Sigma}=Kg_{\Sigma}.

Suppose that $\Sigma$ satisfies $\mathcal{A}^{\prime}$ . It is easy to see that $\mathcal{A}^{\prime}$ is stronger than $\mathcal{A}$ so we can apply Theorem 2.1 which tells us that $H$ is parallel and that

\displaystyle K\mathinner{\!\left\lvert H\right\rvert}^{2}\equiv K\mathinner{\!\left\lvert JH\right\rvert}^{2}\equiv Ric_{\Sigma}(JH,JH)\equiv 0.

Since $H$ is parallel, it has constant norm and therefore $\Sigma$ must be minimal or flat. ∎

Proof of Theorem 3.1.

Let $\Sigma$ be a closed, connected Hamiltonian stationary Lagrangian submanifold of a Kähler-Einstein surface $M$ . Also assume that $\Sigma$ satisfies $\mathcal{A}^{\prime}$ . Then, by Lemma 3.3, $\Sigma$ has parallel mean curvature and it must also be minimal or flat.

First, Suppose that $\Sigma$ is orientable. Then, by the Gauss-Bonnet theorem,

\displaystyle\frac{1}{2\pi}\int_{\Sigma}KdA=\chi(\Sigma)

(4)

where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$ . Since $\chi(\Sigma)=2-2g$ , where $g$ is the genus of $\Sigma$ , and $K$ is non-negative, we see that the genus must be $0$ or $1$ . Therefore, $\Sigma$ is diffeomorphic either to a sphere or to a torus respectively. Equation (4) also tells us that $\Sigma$ is flat if and only if it has genus $1$ , i.e. it is a torus. So, if $\Sigma$ is not flat then it is not just minimal but it must also have genus $0$ and thus it must be a minimal sphere. This completes the proof of the case when $\Sigma$ is orientable.

Now, suppose that $\Sigma$ is not orientable. In this case, $\chi(\Sigma)=2-\hat{g}$ , where $\hat{g}$ is the non-orientable genus of $\Sigma$ which can be defined as the number of copies of $\mathbb{R}P^{2}$ appearing when the surface is represented as a connected sum of projective planes. Also, equation (4) still holds if we interpret the left-hand side as an integral of a density. One can easily see this by passing to the orientable double cover equipped with the pull-back metric. So, similarly to the orientable case, we have that $\hat{g}$ must be $1$ or $2$ and hence $\Sigma$ is diffeomorphic either to a real projective plane or to a Klein bottle respectively. We also see that $\Sigma$ is flat if and only if it is a Klein bottle. Therefore, when $\Sigma$ is not flat, then it is not just minimal but must also have non-orientable genus $1$ and thus it is a minimal real projective plane. This completes the proof of the non-orientable case. ∎

Proof of Theorem 3.2.

It is known that the fundamental group of a non-compact surface is free (see, for example, [17, p. 142]). The first singular homology group of $\Sigma$ with coefficients in $\mathbb{Z}$ is the abelianization of its fundamental group so $H_{1}(\Sigma;\mathbb{Z})$ is the free abelian group on the generator set of $\pi_{1}(\Sigma)$ . Since $H_{1}(\Sigma;\mathbb{Z})$ is free, we know that $H_{1}(\Sigma;\mathbb{Z})=\mathbb{Z}^{b_{1}}$ , where $b_{1}$ is the first Betti number of $\Sigma$ . Therefore, the cardinality of the generator set of $\pi_{1}(\Sigma)$ is equal to $b_{1}$ .

First, assume that $\Sigma$ is orientable. Since $K\geq 0$ , a result of Huber (Theorem 13. in [7]) tells us that $\Sigma$ is finitely connected, i.e. it is homeomorphic to a closed surface with finitely many punctures. Therefore, $b_{1}$ must be finite. Moreover, since the top homology group of a non-compact manifold vanishes identically, we have that $b_{2}=0$ and thus $\chi(\Sigma)=1-b_{1}$ . Also, by Theorem 10. in [7],

\displaystyle\frac{1}{2\pi}\int_{\Sigma}K\leq\chi(\Sigma)

(5)

so we have

b_{1}\leq 1.

If $\Sigma$ is not orientable, then applying the same argument but to the orientable double cover of $\Sigma$ also yields $b_{1}\leq 1$ .

Since the generator set of $\pi_{1}(\Sigma)$ has either $0$ or $1$ element, $\pi_{1}(\Sigma)$ is either trivial or isomorphic to $\mathbb{Z}$ . If $\pi_{1}(\Sigma)=\mathbb{Z}$ , then $\chi(\Sigma)=0$ so, by (5), $\Sigma$ must be flat. If $\Sigma$ is simply connected, then it is diffeomorphic to $\mathbb{R}^{2}$ . Finally, Lemma 3.3 tells us that $\Sigma$ has parallel mean curvature and it is also minimal or flat which finishes the proof. ∎

3.1 Hamiltonian Stationary Lagrangian Surfaces with Non-Negative Gaussian Curvature in Complex Space Forms

Let $M(4c)$ be a complete, connected complex space form of complex dimension $2$ and constant holomorphic sectional curvature $4c$ . Let $\Sigma\subset M(4c)$ be a Lagrangian submanifold.

We have the Wintgen-type inequality (Lemma 2.4. in [13]),

\displaystyle K+\rho_{N}\leq c+\frac{\mathinner{\!\left\lvert H\right\rvert}^{2}}{4}

(6)

where $\rho_{N}\geq 0$ is a normalized (partial) normal scalar curvature²²2A similar inequality can be obtained using the Chen-Ricci inequality presented, for example, in [5]..

First, we look at the case $c=0$ .

Theorem 3.4.

Let $\Sigma\subset M(0)$ be a complete, connected Hamiltonian stationary Lagrangian submanifold. If $\mathcal{A}^{\prime}$ is satisfied, then $\Sigma$ has parallel second fundamental form. Moreover, when the ambient manifold is $\mathbb{C}^{2}$ , then $\Sigma$ is either

•

a Lagrangian plane,
•

a Riemannian product of a circle and a line (a Lagrangian cylinder),
•

or a Riemannian product of two circles (possibly of different radii).

Proof.

Let $\Sigma\subset M(0)$ be a complete, connected Hamiltonian stationary Lagrangian submanifold. Then, by Lemma 3.3, $\Sigma$ has parallel mean curvature and it must also be minimal or flat. Also, by (6),

\displaystyle 0\leq K\leq\frac{\mathinner{\!\left\lvert H\right\rvert}^{2}}{4}

(7)

so we see that if $\Sigma$ is minimal, then it must also be flat. Therefore, we may assume, without a loss of generality, that $\Sigma$ is flat. By Theorem 2.6. in [10, p. 207], $\mathinner{\!\left\lvert\nabla A\right\rvert}=0$ so we can conclude that $\Sigma$ has parallel second fundamental form.

For the rest of the proof, we assume that $M(0)$ is $\mathbb{C}^{2}$ . Let $E_{1},E_{2}$ be a local orthonormal frame on $\Sigma$ . Then $E_{1},E_{2},JE_{1},JE_{2}$ is a local orthonormal frame on $\mathbb{C}^{2}$ and the components $A_{ij}^{k}$ of the second fundamental form $A$ of $\Sigma$ in $\mathbb{C}^{2}$ are given by

A(E_{i},E_{j})=A_{ij}^{k}JE_{k}.

Let $A^{k}$ denote the $2\times 2$ matrix $(A_{ij}^{k})_{i,j}$ . Then, since $\Sigma$ is flat, by Lemma 2.5. in [10, p. 206], its second fundamental form commutes, i.e. $A^{k}A^{l}=A^{l}A^{k}$ for all $k,l=1,2$ . Therefore, by Theorem 2.9. in [10, p. 210], $\Sigma$ is congruent to one of the following standard Lagrangian submanifolds:

1.

$\mathbb{R}^{2}=\{(z_{1},z_{2})\mathrel{\mathop{\mathchar 58\relax}}\text{Im}(z_{1})=0\text{ and Im}(z_{2})=0\}\subset\mathbb{C}^{2}$ (a Lagrangian plane),
2.

$S^{1}(r)\times\mathbb{R}=\{(z_{1},z_{2})\mathrel{\mathop{\mathchar 58\relax}}\mathinner{\!\left\lvert z_{1}\right\rvert}^{2}=r^{2}\text{ and Im}(z_{2})=0\}\subset\mathbb{C}^{2}$ for some $r>0$ (a Lagrangian cylinder),
3.

$S^{1}(r_{1})\times S^{1}(r_{2})=\{(z_{1},z_{2})\mathrel{\mathop{\mathchar 58\relax}}\mathinner{\!\left\lvert z_{1}\right\rvert}^{2}=r_{1}^{2}\text{ and }\mathinner{\!\left\lvert z_{2}\right\rvert}^{2}=r_{2}^{2}\}\subset\mathbb{C}^{2}$ for some $r_{1},r_{2}>0$ (a product of two circles).

∎

Before looking at the cases $c>0$ and $c<0$ , we state some simple corollaries of Theorem 3.4.

Corollary 3.5.

Let $\Sigma\subset\mathbb{C}^{2}$ be a complete, connected Hamiltonian stationary Lagrangian submanifold that has non-negative Gaussian curvature and $\mathinner{\!\left\lvert H\right\rvert}\in L^{p}$ for some $0<p\leq\infty$ . Then it is either a Lagrangian plane, a Lagrangian cylinder or a product of two circles. Moreover, it can only be a cylinder when $p=\infty$ .

Proof.

Let $\Sigma\subset\mathbb{C}^{2}$ be as stated in the corollary. When $p\in(0,\infty)$ , then $\mathcal{A}^{\prime}$ is clearly satisfied. Since $K\geq 0$ , we know by the Bishop-Gromov volume comparison theorem that $Vol(B_{r})\leq\pi r^{2}$ and thus $\Sigma$ has quadratic volume growth. Therefore, as discussed in the previous section after Theorem 2.1, the growth condition (1) is satisfied whenever $\mathinner{\!\left\lvert H\right\rvert}\in L^{\infty}$ . So $\mathcal{A}^{\prime}$ is satisfied when $p=\infty$ as well and we can use Theorem 3.4 to conclude that $\Sigma$ must be a Lagrangian plane, a Lagrangian cylinder or a product of two circles for any $0<p\leq\infty$ .

Finally, we note that when $\Sigma$ is non-compact, then it has infinite volume [21] so it must be minimal if it has a constant mean curvature that is in $L^{p}$ for some $p\in(0,\infty)$ . The standard Lagrangian cylinder in $\mathbb{C}^{2}$ has constant mean curvature but it is neither compact nor minimal so it can only occur when $p=\infty$ . ∎

We say that a complete non-compact submanifold $L$ is asymptoticaly minimal if its mean curvature vector $H$ converges to $0$ at infinty, i.e. $\mathinner{\!\left\lvert H\right\rvert}\rightarrow 0$ as $r(x)\rightarrow\infty$ .

Corollary 3.6.

The only complete, connected, oriented and asymptotically minimal Hamiltonian stationary Lagrangian submanifolds of $\mathbb{C}^{2}$ with non-negative Gaussian curvature are Lagrangian planes.

Since a complete Kähler manifold of positive holomorphic sectional curvature is necessarily simply connected (see, for example, [18]), we may assume that $M(4)$ is $\mathbb{C}\mathbb{P}^{2}$ equipped with the standard Fubini-Study metric which has constant holomorphic sectional curvature $4$ . Let $S^{5}=\{z\in\mathbb{C}^{3}\mathrel{\mathop{\mathchar 58\relax}}\mathinner{\!\left\lvert z\right\rvert}=1\}$ be the unit sphere in $\mathbb{C}^{3}$ equipped with induced metric. Then the map $\Pi\mathrel{\mathop{\mathchar 58\relax}}S^{5}\to\mathbb{C}\mathbb{P}^{2}$ given by $x\mapsto[x]$ , which is usually referred to as the Hopf fibration, can be used to construct Lagrangian immersions into $\mathbb{C}\mathbb{P}^{2}$ . For more details, see, for example, §3. in [3].

Theorem 3.7.

Let $\Sigma\subset\mathbb{C}\mathbb{P}^{2}$ be a closed connected Hamiltonian stationary Lagrangian submanifold with non-negative Gaussian curvature. Then $\Sigma$ is

•

a totally geodesic $\mathbb{R}\mathbb{P}^{2}$ or

•

flat and is locally congruent to the image of $\Pi\circ L$ where $\Pi\mathrel{\mathop{\mathchar 58\relax}}S^{5}\to\mathbb{C}\mathbb{P}^{2}$ is the Hopf fibration and $L\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to S^{5}$ is given by $L(x,y)=(L_{1}(x,y),L_{2}(x,y),L_{3}(x,y))$ with

	$\displaystyle L_{1}(x,y)$	$\displaystyle=\frac{ae^{-i\frac{x}{a}}}{\sqrt{1+a^{2}}},$
	$\displaystyle L_{2}(x,y)$	$\displaystyle=\frac{e^{i(ax+by)}}{\sqrt{1+a^{2}+b^{2}}}\sin\left(\sqrt{1+a^{2}+b^{2}}y\right)\text{ and}$
	$\displaystyle L_{3}(x,y)$	$\displaystyle=\frac{e^{i(ax+by)}}{\sqrt{1+a^{2}}}\left(\cos\left(\sqrt{1+a^{2}+b^{2}}y\right)-\frac{ib}{\sqrt{1+a^{2}+b^{2}}}\sin\left(\sqrt{1+a^{2}+b^{2}}y\right)\right)$

for some real constants $a\neq 0$ and $b$ .

Proof.

Let $\Sigma\subset\mathbb{C}\mathbb{P}^{2}$ be a closed connected Hamiltonian stationary Lagrangian submanifold with non-negative Gaussian curvature. Then $\mathcal{A}^{\prime}$ is satisfied so by Theorem 3.1, $\Sigma$ has parallel mean curvature and it is a minimal sphere, a minimal real projective plane, a flat Klein bottle or a flat torus. If $\Sigma$ is a minimal sphere or a minimal real projective plane then, by Theorem 7. in [20], $\Sigma$ is immersed in such a way that its image is a totally geodesic $\mathbb{R}\mathbb{P}^{2}$ . If $\Sigma$ is a flat Klein bottle or a flat torus then $\Sigma$ has parallel second fundamental form by Theorem 2.6. in [10, p. 207]. Therefore, the result follows from the classification of submanifolds with parallel second fundamental forms in $\mathbb{C}\mathbb{P}^{2}$ given by Theorem 7.1. in [4].

∎

Finally, we consider the case $c<0$ . Let $\mathbb{C}\mathbb{H}^{2}$ denote the complex hyperbolic space of constant holomorphic sectional curvature $-4$ , let $\mathbb{C}^{3}_{1}$ denote $\mathbb{C}^{3}$ equipped with the psuedo-Euclidean metric $g=-dz_{1}d\bar{z}_{1}+dz_{2}d\bar{z}_{2}+dz_{3}d\bar{z}_{3}$ and set $H^{5}_{1}=\{z\in\mathbb{C}^{3}\mathrel{\mathop{\mathchar 58\relax}}g(z,z)=-1\}$ . Then the map $\Pi\mathrel{\mathop{\mathchar 58\relax}}H^{5}_{1}\to\mathbb{C}\mathbb{H}^{2}$ given by $x\mapsto[x]$ , which we will also refer to as the Hopf fibration, can be used to construct Lagrangian immersions into $\mathbb{C}\mathbb{H}^{2}$ . For more details, see, for example, §3. in [3].

Theorem 3.8.

Let $\Sigma\subset M(4c)$ be a complete, connected Hamiltonian stationary Lagrangian submanifold for some $c<0$ . If $\mathcal{A}^{\prime}$ is satisfied, then $\Sigma$ is flat and has parallel second fundamental form. Moreover, when the ambient manifold is $\mathbb{C}\mathbb{H}^{2}$ , then $\Sigma$ is locally congruent to the image of $\Pi\circ L$ where $\Pi\mathrel{\mathop{\mathchar 58\relax}}H^{5}_{1}\to\mathbb{C}\mathbb{H}^{2}$ is the Hopf fibration and $L\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to H^{5}_{1}$ is given by $L(x,y)=(L_{1}(x,y),L_{2}(x,y),L_{3}(x,y))$ where

1.

$L_{1}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{1-a^{2}}}\left(\cosh\left(\sqrt{1-a^{2}-b^{2}}y\right)-\frac{ib}{\sqrt{1-a^{2}-b^{2}}}\sinh\left(\sqrt{1-a^{2}-b^{2}}y\right)\right)$ ,
$L_{2}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{1-a^{2}-b^{2}}}\sinh\left(\sqrt{1-a^{2}-b^{2}}y\right)$ and
$L_{3}(x,y)=\frac{ae^{i\frac{x}{a}}}{\sqrt{1-a^{2}}}$ for some real constants $a$ and $b$ satisfying $a\neq 0$ and $a^{2}+b^{2}<1$ ;
2.

$L_{1}(x,y)=\left(\frac{i}{b}+y\right)e^{i(\sqrt{1-b^{2}}x+by)}$ ,
$L_{2}(x,y)=ye^{i(\sqrt{1-b^{2}}x+by)}$ and
$L_{3}(x,y)=\frac{\sqrt{1-b^{2}}}{b}e^{i\frac{x}{\sqrt{1-b^{2}}}}$ for a real number $0<b^{2}<1$ ;
3.

$L_{1}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{1-a^{2}}}\left(\cos\left(\sqrt{a^{2}+b^{2}-1}y\right)-\frac{ib}{\sqrt{a^{2}+b^{2}-1}}\sin\left(\sqrt{a^{2}+b^{2}-1}y\right)\right)$ ,
$L_{2}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{a^{2}+b^{2}-1}}\sin\left(\sqrt{a^{2}+b^{2}-1}y\right)$ and
$L_{3}(x,y)=\frac{ae^{i\frac{x}{a}}}{\sqrt{1-a^{2}}}$ for some real constants $a$ and $b$ satisfying $0<a^{2}<1$ and $a^{2}+b^{2}>1$ ;
4.

$L_{1}(x,y)=\frac{ae^{i\frac{x}{a}}}{\sqrt{a^{2}-1}}$ ,
$L_{2}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{a^{2}+b^{2}-1}}\sin\left(\sqrt{a^{2}+b^{2}-1}y\right)$ and
$L_{3}(x,y)=\frac{e^{i(ax+by)}}{\sqrt{a^{2}-1}}\left(\cos\left(\sqrt{a^{2}+b^{2}-1}y\right)-\frac{ib}{\sqrt{a^{2}+b^{2}-1}}\sin\left(\sqrt{a^{2}+b^{2}-1}y\right)\right)$ for some real constants $a$ and $b$ satisfying $a^{2}>1$ ;
5.

$L_{1}(x,y)=\frac{e^{ix}}{8b^{2}}\left(i+8b^{2}(i+x)-4by\right)$ ,
$L_{2}(x,y)=\frac{e^{ix}}{8b^{2}}\left(i+8b^{2}x-4by\right)$ and
$L_{3}(x,y)=\frac{e^{i(x+2by)}}{2b}$ for a real number $b\neq 0$ ; or
6.

$L_{1}(x,y)=e^{ix}\left(1+\frac{y^{2}}{2}-ix\right)$ ,
$L_{2}(x,y)=e^{ix}y$ and
$L_{3}(x,y)=e^{ix}\left(\frac{y^{2}}{2}-ix\right)$ .

Proof.

Let $\Sigma\subset M(4c)$ be a complete, connected Hamiltonian stationary Lagrangian submanifold for some $c<0$ . Suppose also that $\mathcal{A}^{\prime}$ is satisfied. Then by Lemma 3.3, $\Sigma$ has parallel mean curvature and it is also flat or minimal. However, since $K\geq 0$ , it is clear from (6) that $\Sigma$ cannot be minimal. Therefore, $\Sigma$ must be flat and thus it has parallel second fundamental form by Theorem 2.6. in [10, p. 207]. When $M(4c)$ is $\mathbb{C}\mathbb{H}^{2}$ , the result follows from the classification of submanifolds with parallel second fundamental forms in $\mathbb{C}\mathbb{H}^{2}$ given by Theorem 7.2. in [4]. ∎

References

Alías et al. [2019] Luis José Alías, Antonio Caminha, and FY do Nascimento. A maximum principle at infinity with applications to geometric vector fields. Journal of Mathematical Analysis and Applications, 474(1):242–247, 2019.
Arsie [2000] Alessandro Arsie. Maslov class and minimality in Calabi–Yau manifolds. Journal of Geometry and Physics, 35(2-3):145–156, 2000.
Chen [1997] Bang-Yen Chen. Interaction of Legendre curves and Lagrangian submanifolds. Israel Journal of Mathematics, 99:69–108, 1997.
Chen et al. [2010] Bang-Yen Chen, Franki Dillen, and Joeri Van der Veken. Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms. International Journal of Mathematics, 21(05):665–686, 2010.
Deng [2009] Shangrong Deng. An improved Chen-Ricci inequality. Int. Electron. J. Geom, 2(2):39–45, 2009.
Grigor’Yan [1999] Alexander Grigor’Yan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bulletin of the American Mathematical Society, 36(2):135–249, 1999.
Huber [1958] Alfred Huber. On subharmonic functions and differential geometry in the large. Commentarii Mathematici Helvetici, 32(1):13–72, 1958.
Innami [1982] Nobuhiro Innami. Splitting theorems of Riemannian manifolds. Compositio Mathematica, 47(3):237–247, 1982.
Karp [1982] Leon Karp. Subharmonic functions on real and complex manifolds. Mathematische Zeitschrift, 179(4):535–554, 1982.
Kon and Yano [1985] Masahiro Kon and Kentaro Yano. Structures on manifolds, volume 3. World scientific, 1985.
Lee [2018] John M Lee. Introduction to Riemannian manifolds, volume 176. Springer, 2018.
Li and Schoen [1984] Peter Li and Richard Schoen. $\textit{L}^{p}$ and mean value properties of subharmonic functions on Riemannian manifolds. Acta Mathematica, 153(1):279–301, 1984.
Mihai [2014] Ion Mihai. On the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms. Nonlinear Analysis: Theory, Methods & Applications, 95:714–720, 2014.
Oh [1993] Yong-Geun Oh. Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Mathematische Zeitschrift, 212(1):175–192, 1993.
Petersen [2006] Peter Petersen. Riemannian geometry, volume 171. Springer, 2006.
Sakai [1996] Takashi Sakai. On Riemannian manifolds admitting a function whose gradient is of constant norm. Kodai Mathematical Journal, 19(1):39–51, 1996.
Stillwell [2012] John Stillwell. Classical topology and combinatorial group theory, volume 72. Springer Science & Business Media, 2012.
Tsukamoto [1957] Yôtarô Tsukamoto. On Kählerian manifolds with positive holomorphic sectional curvature. Proceedings of the Japan Academy, 33(6):333–335, 1957.
Welsh [1986] David J Welsh. On the existence of complete parallel vector fields. Proceedings of the American Mathematical Society, 97(2):311–314, 1986.
Yau [1974] Shing-Tung Yau. Submanifolds with constant mean curvature. American Journal of Mathematics, 96(2):346–366, 1974.
Yau [1976] Shing-Tung Yau. Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana University Mathematics Journal, 25(7):659–670, 1976.