Hypersurfaces with Light-Like Points in a Lorentzian Manifold

Abstract

Consider a constant mean curvature immersion $F:U(\subset \varvec{R}^n)\rightarrow M$ into an arbitrary Lorentzian $(n+1)$-manifold M. A point $o\in U$ is called a light-like point if the first fundamental form $\mathrm{d}s^2$ of F degenerates at o. We denote by $B_F$ the determinant function of the symmetric matrix associated to $\mathrm{d}s^2$ with respect to a local coordinate system at o. A light-like point o is said to be degenerate if the exterior derivative of $B_F$ vanishes at o. We show that if o is a degenerate light-like point, then the image of F contains a light-like geodesic segment of M passing through f(o) (cf. Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz–Minkowski $(n+1)$-space form $\varvec{R}^{n+1}_1$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of F. Several related results are also given.

Appendices

Appendix A: Fermi Coordinate Systems Along Light-Like Geodesics

Let (M, g) be a Lorentzian $(n+1)$-manifold. In this appendix, we prove the following assertion:

Proposition A.1

Let $I=(a,b)$ be a closed interval, and let $\sigma :I\rightarrow M$ be a light-like geodesic. Then there exists a local diffeomorphism $(\epsilon >0$ is a constant)

$$\begin{aligned}&\Phi :\biggl \{(x_0,\dots ,x_n)\in \varvec{R}^{n+1}\,;\, x_0,x_n\in \left[ \frac{a+\epsilon }{\sqrt{2}},\frac{b-\epsilon }{\sqrt{2}}\right] , \\&\qquad \qquad \,\, |x_i|<\epsilon \,\, (i=1,\dots ,n-1) \biggr \} \rightarrow M \end{aligned}$$

such that

• (a1) $\Phi (t,0,\dots ,t)=\sigma (t)$ holds for $t\in \left[ {a}/{\sqrt{2}},{b}/{\sqrt{2}}\right]$,

• (a2) regarding $(x_0,\dots ,x_n)$ as a local coordinate system by $\Phi$, $g_{0,0}^{}=-1$,   $g_{0,i}^{}=0$ along $\sigma$ and $g_{j,k}^{}=\delta _{j,k}$ hold for $i,j,k=1,\dots ,n$ along $\sigma$, where $g_{j,k}^{}:=g(\partial _{x_j},\partial _{x_k})$, and

• (a3) all of the Christoffel symbols vanish along $\sigma$. As a consequence, all derivatives $\partial g_{\alpha ,\beta }^{}/\partial x_\gamma$$(\alpha ,\beta ,\gamma =0,\dots ,n)$ vanish along $\sigma$.

Proof

We can take vectors $\varvec{e}_0,\dots ,\varvec{e}_n\in T_{\sigma (a)}M$ such that

• (1) $\varvec{e}_0,\varvec{e}_1,\dots ,\varvec{e}_n$ ($\varvec{e}_0:=\sigma '(a)$) gives a basis of $T_{\sigma (a)} M$,

• (2) $g(\varvec{e}_0,\varvec{e}_0)= g(\varvec{e}_1,\varvec{e}_1)=0$ and $g(\varvec{e}_0,\varvec{e}_1)=-1$,

• (3) $g(\varvec{e}_0,\varvec{e}_i)=g(\varvec{e}_1,\varvec{e}_i)=0$ for $i=2,\dots ,n$,

• (4) $g(\varvec{e}_j,\varvec{e}_k)=\delta _{j,k}$ for $j,k=2,\dots ,n$.

We let $E_\alpha (t)$ ($a\le t\le b$,   $\alpha =0,\dots ,n$) be the parallel vector field along $\sigma (t)$ such that

$$\begin{aligned} E_\alpha (a)=\varvec{e}_\alpha \qquad (\alpha =0,\dots ,n). \end{aligned}$$

Since $\sigma$ is a geodesic, $\sigma '(t)=E_0(t)$ holds for $t\in I$. We then set

$$\begin{aligned} \Phi :\varvec{R}\times \varvec{R}^n\ni (t,y_1,\dots ,y_n)\mapsto {{\text {Exp}}}_{\sigma (t)}\left( y_1E_1(t)+\cdots +y_n E_n(t)\right) \in M, \end{aligned}$$

where ${{\text {Exp}}}_p$ is the exponential map at $p\in M$ with respect to the Lorentzian metric g. Then there exists $\epsilon >0$ such that the restriction of $\Phi$ to the open domain

$$\begin{aligned} \left\{ (y_0,y_1\dots ,y_n)\in \varvec{R}\times \varvec{R}^n;\, a<y_0<b,\quad |y_i|<\epsilon \,\,\, (i=1,\dots ,n)\right\} \end{aligned}$$

gives a diffeomorphism, where $y_0:=t$. By definition, we have

$$\begin{aligned} E_{\alpha }(t) =\left( \frac{\partial }{\partial y_{\alpha }}\right) _{\sigma (t)}\qquad (\alpha =0,\dots ,n), \end{aligned}$$

and

• $g(\partial _{y_0},\partial _{y_0})= g(\partial _{y_1},\partial _{y_1})=0$ and $g(\partial _{y_0},\partial _{y_1})=-1$,

• $g(\partial _{y_0},\partial _{y_i})=g(\partial _{y_1}, \partial _{y_i})=0$ for $i=1,\dots ,n$ holds along $\sigma$,

• $g(\partial _{y_j},\partial _{y_k})=\delta _{j,k}$ for $j,k=2,\dots ,n$ along $\sigma$.

Let D be the Levi-Civita connection of M. Then the Christoffel symbols with respect to this local coordinate system $(y_0,\dots ,y_n)$ are defined by

$$\begin{aligned} D_{\partial _{y_\alpha }}\partial _{y_\beta }= \sum _{\gamma =0}^n\Gamma _{\alpha ,\beta }^\gamma \partial _{y_\gamma } \qquad (\alpha ,\beta =0,\dots ,n). \end{aligned}$$

We would like to show that all of the Christoffel symbols $\Gamma _{\alpha ,\beta }^\gamma$ ($\alpha ,\beta ,\gamma =0,\dots ,n$) with respect to the local coordinates $(y_0,\dots ,y_n)$ vanish along the $y_0$-axis (i.e., along the curve $\sigma$). We fix $(a_1,\dots ,a_n)\in \varvec{R}^n\setminus \{\varvec{0}\}$, and consider a curve $c(t):=(0,a_1t,\dots ,a_n t)$ in M. Then, by our definition of the local coordinate system $(y_0,\dots ,y_n)$, this curve c(t) gives a geodesic on M. So $c(t)=(c_0(t),\dots ,c_n(t))$ satisfies

$$\begin{aligned} c''_\gamma (t)+\sum _{\alpha ,\beta =0}^n \Gamma _{\alpha ,\beta }^\gamma (c(t)) c'_\alpha (t) c'_\beta (t)=0 \qquad (\gamma =0,\dots ,n). \end{aligned}$$

Since $c'_0(t)=c''_0(t)=\cdots =c''_n(t)=0$, and $c'_i(t)=a_i$ for $i=1,\dots ,n$, this reduces to

$$\begin{aligned} \sum _{i,j=1}^n \Gamma _{i,j}^\gamma (c(t))a_i a_j=0 \qquad (\gamma =0,\dots ,n). \end{aligned}$$

If we set $a_i=a_j=1$ and other $a_k=0$ for $k\ne i,j$, then we have

$$\begin{aligned} \Gamma _{i,j}^\gamma (c(t))+\Gamma _{j,i}^\gamma (c(t))=0 \qquad (i,j=1,\dots ,n,\,\,\gamma =0,\dots ,n). \end{aligned}$$

Since the connection is torsion free, $\Gamma _{\alpha ,\beta }^\gamma =\Gamma _{\beta ,\alpha }^\gamma$ holds ($0\le \alpha ,\beta ,\gamma \le n$), and we get

$$\begin{aligned} \Gamma _{i,j}^\gamma (c(t))=0 \qquad (i,j=1,\dots ,n,\,\,\gamma =0,\dots ,n). \end{aligned}$$

(A.1)

On the other hand, since $\sigma$ is a geodesic and $E_0(t),\dots , E_n(t)$ are parallel vector fields along $\sigma$, we have

$$\begin{aligned} \Gamma _{0,\beta }^\gamma (c(t))=0 \qquad (\beta ,\gamma =0,\dots ,n). \end{aligned}$$

(A.2)

By (A.1) and (A.2) together with the property $\Gamma _{\alpha ,\beta }^\gamma =\Gamma _{\beta ,\alpha }^\gamma$, we can conclude that all of the Christoffel symbols along the curve $\sigma$ vanish.

We now set

$$\begin{aligned} x_0:=\frac{y_0+y_1}{\sqrt{2}},\quad x_n:=\frac{y_0-y_1}{\sqrt{2}}, \quad x_i:=y_{i+1} \qquad (i=1,\dots ,n-1). \end{aligned}$$

(A.3)

Then the properties (a1) and (a2) for this new coordinate system $(x_0,\dots ,x_n)$ are obvious. Since the property that all of the Christoffel symbols vanish along $\sigma$ is preserved under linear coordinate changes, (a3) is also obtained. $\square$

Appendix B: Computations in $\varvec{R}^{n+1}_1$

We denote by the dot ‘$\cdot$’ the canonical Lorentzian inner product of $\varvec{R}^{n+1}_1$ with signature $(-+\cdots +)$. In this appendix, we compute $B:=B_F$ and $A:=A_F$ with respect to the canonical coordinate system $(x_0,x_1,\dots ,x_n)$ of $\varvec{R}^{n+1}_1$. Let $f(x_1,\dots ,x_n)$ be a $C^2$-function of n variables defined on a neighborhood of the origin $o\in \varvec{R}^n$. We set

$$\begin{aligned} F=(f(x_1,\dots ,x_n),x_1,\dots ,x_n) \end{aligned}$$

and

$$\begin{aligned} \varvec{e}_0:=(1,0,\dots ,0),\quad \varvec{e}_1:=(0,1,\dots ,0),\quad \dots ,\quad \varvec{e}_n:=(0,0,\dots ,1). \end{aligned}$$

They give a canonical frame in $\varvec{R}^{n+1}_1$ satisfying $\varvec{e}_0\cdot \varvec{e}_0=-1$. Then $(x_1,\dots ,x_n)$ gives a local coordinate system of the domain of F, and we have

$$\begin{aligned} F_{x_i}=f_{x_i}\varvec{e}_0+\varvec{e}_i \qquad (i=1,\dots ,n). \end{aligned}$$

We set

$$\begin{aligned} S:=(s_{i,j}^{})_{i,j=1,\dots ,n},\qquad s_{i,j}^{}:=F_{x_i}\cdot F_{x_j} \end{aligned}$$

for $i,j=1,\dots ,n$, where the dot ‘$\cdot$ ’ is the canonical inner product of $\varvec{R}^{n+1}_1$. Then we have

$$\begin{aligned} s_{i,j}^{}=\delta _{i,j}-f_{x_i}f_{x_j}\qquad \qquad (i,j=1,\dots ,n), \end{aligned}$$

(B.1)

and

$$\begin{aligned} S=I_n-(\nabla f)^T(\nabla f), \qquad \nabla f:=(f_{x_1},\dots ,f_{x_n}) \end{aligned}$$

hold, where $I_n$ is the identity matrix and $(\nabla f)^T$ is the transpose of $(\nabla f)$. Then we have

$$\begin{aligned} B={{\text {det}}}(S)=(1-\lambda _1)\cdots (1-\lambda _n), \end{aligned}$$

where $\lambda _1,\dots ,\lambda _n$ are eigenvalues of the matrix $(\nabla f)^T(\nabla f)$. Since $(\nabla f)^T(\nabla f)$ is of rank 1, we may assume that $\lambda _2=\cdots =\lambda _n=0$. Then we have

$$\begin{aligned} B=1-\lambda _1=1-{{\text {trace}}}((\nabla f)^T(\nabla f)) =1-(f_{x_1})^2-\cdots -(f_{x_n})^2. \end{aligned}$$

(B.2)

Using this, it can be easily checked that the inverse matrix of S is given by

$$\begin{aligned} S^{-1}=I_n+\frac{1}{B}(\nabla f)^T(\nabla f). \end{aligned}$$

In particular, the cofactor matrix $\tilde{S}=({\tilde{s}}^{i,j})_{i,j=1,\dots ,n}$ of S satisfies

$$\begin{aligned} \tilde{S}=B I_n+(\nabla f)^T(\nabla f), \end{aligned}$$

(B.3)

that is

$$\begin{aligned} {\tilde{s}}^{i,j}=B\delta _{i,j}+f_{x_i}f_{x_j}\qquad \qquad (i,j=1,\dots ,n), \end{aligned}$$

(B.4)

where $\delta _{i,j}$ denotes Kronecker’s delta.

On the other hand,

$$\begin{aligned} {\tilde{\nu }}=-(1,f_{x_1},\dots ,f_{x_n}) \end{aligned}$$

(B.5)

gives a normal vector field defined by (2.5). Then the coefficients ${\tilde{h}}_{i,j}$ of the normalized second fundamental form given in (2.6) are written as

$$\begin{aligned} {\tilde{h}}_{i,j}=F_{x_i,x_j}\cdot {\tilde{\nu }}=f_{x_i,x_j}. \end{aligned}$$

(B.6)

Thus the matrix ${\tilde{h}}:=({\tilde{h}}_{i,j})_{i,j=1,\dots ,n}$ is just the Hessian matrix of f. By using the identity (B.2), the function A given in (2.6) can be computed as follows:

$$\begin{aligned} A&={{\text {trace}}}(\tilde{S} {\tilde{h}})= \sum _{i,j=1}^n (B\delta _{i,j}+f_{x_i}f_{x_j})f_{x_i,x_j}\\&=B \sum _{i=1}^n \left( f_{x_i,x_i} -\frac{1}{2} B_{x_i}f_{x_i}\right) =B \triangle f -\frac{1}{2} \nabla B\star \nabla f, \end{aligned}$$

where ‘$\star$’ is the canonical inner product of $\varvec{R}^n$.