Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications

Abstract

With several concrete examples of zero mean curvature surfaces in the Lorentz-Minkowski 3-space $\varvec{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first fundamental form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in $\varvec{R}^3_1$ whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.

Umehara was partially supported by the Grant-in-Aid for Scientific Research (A) No. 26247005, and Yamada by (C) No. 26400066 from Japan Society for the Promotion of Science.

Appendix A. Division Lemma

Lemma A.1

Let g be a $C^r$-function ($r\ge 1$) defined on a convex domain U of the xy-plane including the origin o, satisfying

$$\begin{aligned} g(0,y)=\frac{\partial g}{\partial x}(0,y) =\frac{\partial ^2 g}{\partial x^2}(0,y) =\dots =\frac{\partial ^k g}{\partial x^k}(0,y)=0\qquad \bigl ((0,y)\in U\bigr ) \end{aligned}$$

(A.1)

for a nonnegative integer $k<r$. Then there exists a $C^{r-k-1}$-function h defined on U such that

$$\begin{aligned} g(x,y)=x^{k+1}h(x,y)\qquad \bigl ((x,y)\in U\bigr ). \end{aligned}$$

(A.2)

Proof

We shall prove by an induction in k. Since

$$\begin{aligned} g(x,y)&= \int _0^1 \frac{d g(tx,y)}{d t}\,dt = \int _0^1 x g_x(tx,y) dt=x \int _0^1 g_x(tx,y)\, dt, \end{aligned}$$

the conclusion follows for $k=0$, by setting

$$ h(x,y):=\int _0^1 g_x(tx,y)\, dt. $$

Assume that the statement holds for $k-1$. If g satisfies (A.1), there exists a $C^{r-k}$-function $\varphi (x,y)$ defined on U such that

$$\begin{aligned} g(x,y)=x^k \varphi (x,y)\qquad \bigl ((x,y)\in U\bigr ). \end{aligned}$$

(A.3)

Differentiating this k-times in x, we have

$$ 0=\frac{\partial ^k g}{\partial x^k}(0,y) = k! \varphi (0,y) $$

because of (A.1). Hence, by the case $k=0$ of this lemma, there exists $C^{k-r-1}$-function h(x, y) defined on U such that $\varphi (x,y)=x h(x,y)$. The function h is the desired one. $\square$