Evolution of spacelike surfaces in $\mathrm{AdS}_3$ by their Lagrangian angle

Abstract

We study spacelike hypersurfaces $M$ in an anti-De Sitter spacetime $N$ of constant sectional curvature $-\kappa , \kappa >0$ that evolve by the Lagrangian angle of their Gauß maps. In the two dimensional case we prove a convergence result to a maximal spacelike surface, if the Gauß curvature $K$ of the initial surface $M\subset N$ and the sectional curvature of $N$ satisfy $|K|<\kappa$.

Appendix

With the same notations as before let us now assume that $F:M\rightarrow N$ is a spacelike immersion of a surface $M$ into the anti-De Sitter manifold $N=\mathrm{AdS}_3$, represented by

$$\begin{aligned} \mathrm{AdS}_3=\{V\in \mathbb R _2^4:\left\langle {V},{V}\right\rangle _{~_{2,4}}=-1\} \end{aligned}$$

with its induced Lorentzian metric $\left\langle {\cdot },{\cdot }\right\rangle _{~_{N}}$ that it inherits from $\mathbb R _2^4=({\mathbb R ^{4}},\left\langle {\cdot },{\cdot }\right\rangle _{~_{2,4}})$, where the inner product on $\mathbb R _2^4$ is given by

$$\begin{aligned} \left\langle {V},{W}\right\rangle _{~_{2,4}}=V^1W^1+V^2W^2-V^3W^3-V^4W^4. \end{aligned}$$

$\mathrm{AdS}_3$ is a $3$-dimensional Lorentzian space form of constant sectional curvature $-1$ and is a vacuum solution of Einstein’s equation with cosmological constant $\varLambda <0$. $\mathrm{AdS}_3$ contains closed timelike curves and hence is not simply connected. The simply connected universal cover of $\mathrm{AdS}_3$ will also be called anti-De Sitter space and we denote it $\widetilde{\mathrm{AdS}_3}$.

Thus an immersion $F:M\rightarrow \mathrm{AdS}_3$ can also be seen as an immersion of $M$ into $\mathbb R _2^4$. The Gauß map of $F$ is the map

$$\begin{aligned} \fancyscript{G}:M\rightarrow Gr_2^+(2,4),\quad p\mapsto T_pM\in Gr_2^+(2,4), \end{aligned}$$

where $T_pM$ is considered as an oriented spacelike surface in $\mathbb R _2^4$ and $Gr_2^+(2,4)$ denotes the Grassmannian of oriented spacelike surfaces in $\mathbb R _2^4$. It is well known that the Grassmannian $Gr_2^+(2,4)$ is isometric to $\mathbb H _{1/\sqrt{2}}\times \mathbb H _{1/\sqrt{2}}$, where $\mathbb H _{1/\sqrt{2}}$ denotes the scaled hyperbolic plane

$$\begin{aligned} \mathbb H _{1/\sqrt{2}}:=\left\{ V\in \mathbb R _1^3:\left\langle {V},{V}\right\rangle _{~_{1,3}}=-\frac{1}{2}, V^3>0\right\} , \end{aligned}$$

where $(\mathbb R _1^3,\left\langle {\cdot },{\cdot }\right\rangle _{~_{1,3}})$ denotes the usual Minkowski space. To understand the geometry of the Gauß map $\fancyscript{G}$ it is convenient to use the isometry between $Gr_2^+(2,4)$ and $\mathbb H _{1/\sqrt{2}}\times \mathbb H _{1/\sqrt{2}}$ (since the sectional curvature of $\mathbb H _{1/\sqrt{2}}$ is $-2, Gr_2^+(2,4)$ is a Kähler–Einstein manifold of scalar curvature $S=-8$).

Let $e_1,e_2,e_3,e_4$ denote the standard basis of ${\mathbb R ^{4}}$. We introduce a set of endomorphisms on ${\mathbb R ^{4}}$:

$$\begin{aligned} E_+^1:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} e_4\\ -e_3\\ -e_2\\ e_1 \end{array}\right),&\quad E_-^1:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} -e_4\\ -e_3\\ -e_2\\ -e_1 \end{array}\right), \nonumber \\ E_+^2:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} -e_3\\ -e_4\\ -e_1\\ -e_2 \end{array}\right),&\quad E_-^2:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} -e_3\\ e_4\\ -e_1\\ e_2 \end{array}\right),\\ E_+^3:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} e_2\\ -e_1\\ -e_4\\ e_3 \end{array}\right),&\quad E_-^3:\, \left(\begin{array}{l} e_1\\ e_2\\ e_3\\ e_4 \end{array}\right) \mapsto \left(\begin{array}{l} e_2\\ -e_1\\ e_4\\ -e_3 \end{array}\right),\nonumber \end{aligned}$$

(46)

These endomorphisms satisfy

$$\begin{aligned}&E_+^1E_+^2=-E^2_+E^1_+=E^3_+,\quad (E^1_+)^2=(E^2_+)^2=-(E^3_+)^2=\mathrm{Id},\\&E_-^1E_-^2=-E^2_-E^1_-=E^3_-,\quad (E^1_-)^2=(E^2_-)^2=-(E^3_-)^2=\mathrm{Id} \end{aligned}$$

Moreover, if $V\in {\mathbb R ^{4}}$ is an arbitrary nonzero vector, then

$$\begin{aligned} \{V,E_+^1V,E_+^2V,E_+^3V\} \end{aligned}$$

forms a positively oriented basis and

$$\begin{aligned} \{V,E_-^1V,E_-^2V,E_-^3V\} \end{aligned}$$

a negatively oriented basis of ${\mathbb R ^{4}}$. Associated to these endomorphisms are the following six symplectic $2$-forms:

$$\begin{aligned} \omega ^A_+:=\left\langle {E^A_+\cdot },{\cdot }\right\rangle _{~_{2,4}}\quad \text{ and}\quad \omega ^A_- :=\left\langle {E^A_-\cdot },{\cdot }\right\rangle _{~_{2,4}}\!\!,\qquad A=1,2,3 \end{aligned}$$

and we have

$$\begin{aligned} \omega ^1_+&= e_2\wedge e_3+e_4\wedge e_1,\quad \omega ^1_-=e_2\wedge e_3-e_4\wedge e_1,\\ \omega ^2_+&= e_1\wedge e_3+e_2\wedge e_4,\quad \omega ^2_-=e_1\wedge e_3-e_2\wedge e_4,\\ \omega ^3_+&= e_1\wedge e_2+e_3\wedge e_4,\quad \omega ^3_-=e_1\wedge e_2-e_3\wedge e_4. \end{aligned}$$

For any spacelike unit vector $e$ and any two vectors $V,W$ we have

$$\begin{aligned} \left\langle {V},{W}\right\rangle _{~_{2,4}}&= -\omega ^1_+(e,V)\omega ^1_+(e,W)-\omega ^2_+(e,V) \omega ^2_+(e,W)\nonumber \\&+\,\omega ^3_+(e,V)\omega ^3_+(e,W)+\left\langle {e},{V}\right\rangle _{~_{2,4}}\left\langle {e},{W}\right\rangle _{~_{2,4}}\nonumber \\&= -\omega ^1_-(e,V)\omega ^1_-(e,W)-\omega ^2_-(e,V)\omega ^2_-(e,W) \nonumber \\&+\,\omega ^3_-(e,V)\omega ^3_-(e,W)+\left\langle {e},{V}\right\rangle _{~_{2,4}}\left\langle {e},{W}\right\rangle _{~_{2,4}}\!\!. \end{aligned}$$

(47)

Let $\nu _N$ denote the future directed timelike unit normal along $\mathrm{AdS}_3\subset \mathbb R _2^4$. The orientation of ${\mathbb R ^{4}}$ induces an orientation on $\mathrm{AdS}_3$ in the following way: We say that $V_1,V_2,V_3\!\in \! T_q\mathrm{AdS}_3$ is positively oriented, if $V_1,V_2,V_3,\nu _N$ represents the positive orientation of ${\mathbb R ^{4}}$.

Now let $F:M\rightarrow \mathrm{AdS}_3$ be a spacelike immersion of an oriented surface and let $\nu$ denote the future directed timelike unit normal of $M$ within $\mathrm{AdS}_3$. We will assume that the orientation of $M$ is chosen in such a way that for any positively oriented basis $\{e_1,e_2\}$ of $T_pM$ the basis $\{DF(e_1),DF(e_2),\nu \}$ represents the positive orientation of $T_{F(p)}\mathrm{AdS}_3$.

For such an immersion let us define the six functions

$$\begin{aligned} \fancyscript{G}_+^A:M\rightarrow {\mathbb R ^{}},\quad \fancyscript{G}_+^A:=\frac{1}{\sqrt{2}}*(F^*\omega _+^A),\quad A=1,2,3 \end{aligned}$$

and

$$\begin{aligned} \fancyscript{G}_-^A:M\rightarrow {\mathbb R ^{}},\quad \fancyscript{G}_-^A:=\frac{1}{\sqrt{2}}*(F^*\omega _-^A) ,\quad A=1,2,3, \end{aligned}$$

where $F^*$ denotes “pull-back” and $*$ is the Hodge-Operator on forms. From Eq. (47) one immediately gets¹

$$\begin{aligned} (\fancyscript{G}_+^1)^2+(\fancyscript{G}_+^2)^2-(\fancyscript{G}_+^3)^2 =-\frac{1}{2}=(\fancyscript{G}_-^1)^2+(\fancyscript{G}_-^2)^2 -(\fancyscript{G}_-^3)^2 \end{aligned}$$

(48)

and by construction we have $\fancyscript{G}^3_\pm >0$, so that

$$\begin{aligned} \fancyscript{G}_+=(\fancyscript{G}_+^1,\fancyscript{G}_+^2,\fancyscript{G}_+^3) \quad \text{ and}\quad \fancyscript{G}_-=(\fancyscript{G}_-^1,\fancyscript{G}_-^2,\fancyscript{G}_-^3) \end{aligned}$$

define two functions from $M$ to $\mathbb H _{1/\sqrt{2}}$. $\fancyscript{G}_+,\fancyscript{G}_-$ are called the self-dual resp. the anti-self-dual Gauß maps of $F$ and the Gauß map $\fancyscript{G}:M\rightarrow Gr^+_2(2,4)=\mathbb H _{1/\sqrt{2}}\times \mathbb H _{1/\sqrt{2}}$ is given by the pair $\fancyscript{G}=(\fancyscript{G}_+,\fancyscript{G}_-)$.

Let

$$\begin{aligned} \langle \cdot ,\cdot \rangle _{~_{ Gr_2^+(2,4)}},\quad \fancyscript{J}\quad \text{ and}\quad \omega =\langle \fancyscript{J}\cdot ,\cdot \rangle _{~_{ Gr_2^+(2,4)}} \end{aligned}$$

denote the Kähler metric, complex structure and Kähler form on the Grassmannian $Gr_2^+(2,4)=\mathbb H _{1/\sqrt{2}}\times \mathbb H _{1/\sqrt{2}}$.

As was shown in [18, 19], we have $\fancyscript{G}^*\omega =0$, i.e. the Gauß map of an immersion $F:M\rightarrow \mathrm{AdS}_3$ defines a Lagrangian immersion $\fancyscript{G}:M\rightarrow Gr_2^+(2,4)$.

Let

$$\begin{aligned} {\sigma }_{ij} dx^i\otimes dx^j=\fancyscript{G}^*\langle \cdot ,\cdot \rangle _{~_{ Gr_2^+(2,4)}} \end{aligned}$$

denote the Riemannian metric on $M$ induced by the Gauß map. If $D$ denotes the connection associated to $\sigma$, then it is well known that the second fundamental tensor

$$\begin{aligned} {\tau }_{ijk}=\omega (D_i\fancyscript{G},D_jD_k\fancyscript{G}) \end{aligned}$$

of the Lagrangian immersion is completely symmetric and that the mean curvature form $\tau =\tau _idx^i$ on $M$, i.e. its trace $\tau _i={\sigma }^{jk}{\tau }_{ijk}$ (where $({\sigma }^{jk})_{j,k=1,2}$ denotes the inverse of $({\sigma }_{jk})_{j,k=1,2}$), is closed.

The first and second fundamental forms on $M$ induced by $F$ shall be denoted (as before) by ${g}_{ij} dx^i\otimes dx^j$ and ${h}_{ij} dx^i\otimes dx^j$. If we consider $F$ as a map from $M$ to $\mathbb R _2^4$, then the Gauß formula shows that the second fundamental tensor $\tilde{A}$ of $M$, considered as a submanifold of codimension two in $\mathbb R _2^4$, decomposes into

$$\begin{aligned} {\tilde{A}}_{ij}={g}_{ij}\nu _N+{h}_{ij}\nu . \end{aligned}$$

(49)

 

Lemma 12

Let $F:M\rightarrow \mathrm{AdS}_3$ be a spacelike immersion. With the same notations as above the following relations between the first and second fundamental forms of $F$ and $\fancyscript{G}$ are valid:

$$\begin{aligned} {\sigma }_{ij}&= {g}_{ij}+{g}^{kl}{h}_{ik}{h}_{jl},\end{aligned}$$

(50)

$$\begin{aligned} {\tau }_{ijk}&= \nabla _i{h}_{jk}, \end{aligned}$$

(51)

where $\nabla$ denotes the Levi-Civita connection of ${g}_{ij}$.

 

Proof

Straightforward computations using (47), (48) and (49).$\square$

 

In particular, we observe that ${\sigma }_{ij}$ coincides with the tensor defined earlier in Eq. (18) since in this special situation we have $\kappa =1$. As a corollary we obtain:

 

Lemma 13

The Maslov class of the Gauß map $\fancyscript{G}$ is trivial and the Lagrangian angle is given by $\phi =\arctan \lambda _1+\arctan \lambda _2$, where $\lambda _1,\lambda _2$ are the principal curvatures of $F:M\rightarrow \mathrm{AdS}_3$.

 

Proof

We have seen in Lemma 3 that $\phi =\arctan \lambda _1+\arctan \lambda _2$ is a smooth function. Moreover we have $\frac{\partial \phi }{\partial {h}_{ij}}={\sigma }^{ij}$ and then

$$\begin{aligned} \nabla _k\phi&= \frac{\partial \phi }{\partial {h}_{ij}}\nabla _k{h}_{ij}+\frac{\partial \phi }{\partial {g}^{ij}}\nabla _k{g}^{ij}\\&= {\sigma }^{ij}\nabla _k{h}_{ij}\\&\overset{(51)}{=}{\sigma }^{ij}{\tau }_{kij}\\&= \tau _k. \end{aligned}$$

This means that the mean curvature form $\tau$ of the Gauß map satisfies $\tau =d\phi$. Since $\tau /\pi$ represents the Maslov class, it must be trivial.$\square$

We will now treat the case where $F:M\times [0,T)\rightarrow \mathrm{AdS}_3$ is a smooth family of spacelike immersions satisfying an evolution equation of the form

$$\begin{aligned} \frac{d}{dt}\,F=f\nu , \end{aligned}$$

where $f$ is an arbitrary smooth function and $\nu$ the future directed timelike unit normal. The Gauß maps of $F$ depend on $t$ and will vary in time. A straightforward computation gives the two relations

$$\begin{aligned} \left\langle \frac{d}{dt}\,\fancyscript{G},D_k\fancyscript{G}\right\rangle _{~_{ Gr_2^+(2,4)}} ={g}^{ml}{h}_{lk}\nabla _mf \end{aligned}$$

and

$$\begin{aligned} \left\langle \frac{d}{dt}\,\fancyscript{G},\fancyscript{J}D_k\fancyscript{G}\right\rangle _{~_{ Gr_2^+(2,4)}} =\nabla _kf, \end{aligned}$$

so that

$$\begin{aligned} \frac{d}{dt}\,\fancyscript{G}=\fancyscript{J}({\sigma }^{kl}\nabla _kfD_l\fancyscript{G})+{\sigma }^{kl}{g}^{ms}{h}_{sk}D_mfD_l\fancyscript{G}. \end{aligned}$$

So we have shown:

 

Lemma 14

Suppose $F:M\times [0,T)\rightarrow \mathrm{AdS}_3$ is a smooth family of spacelike immersions driven by the flow $\frac{d}{dt}\,F=f\nu$, where $\nu$ denotes the future directed timelike unit normal. Then the Gauß maps $\fancyscript{G}_F:M\times [0,T)\rightarrow Gr_2^+(2,4)$ of $F$ evolve according to

$$\begin{aligned} \frac{d}{dt}\,\fancyscript{G}_F=\left(\fancyscript{J}\circ d\fancyscript{G}_F+d\fancyscript{G}_F\circ \fancyscript{W}_F\right)\nabla ^\sigma f. \end{aligned}$$

(52)

where$\fancyscript{J}$ denotes the complex structure on $Gr_2^+(2,4), \fancyscript{W}_F$ is the Weingarten map of $F$ and $\nabla ^\sigma f$ denotes the gradient of $f$ w.r.t. the induced metric $\sigma =\fancyscript{G}^*\left\langle \cdot ,\cdot \right\rangle _{~_{ Gr_2^+(2,4)}}$.

In particular, if we choose for $f$ the Lagrangian angle $\phi =\arctan \lambda _1+\arctan \lambda _2$, then—up to the tangential term $(d\fancyscript{G}_F\circ \fancyscript{W}_F)\nabla ^\sigma f$, which is of no interest concerning the geometric evolution—the Gauß maps evolve by the Lagrangian mean curvature flow.