Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

By S. Akamine, M. Umehara, K. Yamada

Abstract

Calabi’s Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space and maximal surfaces in Lorentz-Minkowski space , we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a plane.

1. Introduction

Consider a -dimensional barotropic steady flow on a simply connected domain in the -plane whose velocity vector field is , with density and pressure . We assume there are no external forces. Then

the flow is a foliation of the integral curve of ,

is a scalar field on ,

is a monotone function of ,

() is called the local speed of sound.

The following Euler’s equation of motion holds:

We also assume the flow is irrotational; that is,

where Here, ‘the equation of continuity’ is equivalent to the fact that

By Equation 1.2, there exists a function , called the potential of the flow, such that , where . Since is a function of , the fact and Equation 1.1 yield that

By Equation 1.3, one can easily check that

On the other hand, by Equation 1.3, there exists a function , called the stream function of the flow, such that

If we set and , Equation 1.4 can be written as

Since

the identity yields that

A flow satisfying

is called a Chaplygin gas flow (see Reference 4, p. 24 and also Reference 11, Section 4). For a given stream function of the Chaplygin gas flow, we set

Let be a domain in the -plane . Let be an immersion into the Lorentz-Minkowski 3-space of signature . We set

and

where denotes the canonical Lorentzian inner product of and denotes the determinant of the matrix . A point where (resp., , ) is said to be space-like (resp., time-like, light-like). We set

where and is the canonical Lorentzian vector product of . Consider the matrix and set

where is the cofactor matrix of . We call a zero mean curvature surface if vanishes identically. In this paper, for the sake of simplicity, we abbreviate ‘zero mean curvature’ by ‘ZMC’. A ZMC-surface consisting only of space-like points is called a maximal surface. On the other hand, a surface in consisting only of light-like points is called a light-like surface. It is known that the identity implies that (see Reference 21, Proposition 2.1). In particular, any light-like surfaces are ZMC-surfaces in our sense. Moreover, at a point where , the mean curvature function of is well-defined, and is equivalent to the condition that .

We now assume that is written in the form . Then it can be easily checked that (cf. Equation 1.9) and

Under the condition Equation 1.8, the equation Equation 1.7 for the stream function reduces to

which implies that vanishes identically. So we call this the ZMC-equation in . If , then we have ; that is, is obtained. Substituting this into Equation 1.1, we get , and so there exists a constant such that

By Equation 1.6, we can rewrite this as

By Equation 1.11 and Equation 1.12, the sign change of corresponds to the type change of the Chaplygin gas flow from sub-sonic () to super-sonic (); that is, the sub-sonic part satisfies . If , then vanishes identically, and the graph of gives a light-like surface. Such surfaces are discussed in the appendix, and we now consider the case . Since and have the same sign (cf. Equation 1.12), we can write

By Equation 1.11 and the fact that , Equation 1.5 can be written as

We set

If , then Equation 1.14 reduces to

which is known as the condition that the graph of gives a minimal surface in the Euclidean 3-space . On the other hand, if , then Equation 1.14 reduces to

which is the ZMC-equation (cf. Equation 1.10). It can be easily checked that the graph of is a time-like ZMC-surface in . In both of the two cases, it can be easily checked that ()

holds, where Note that satisfies Equation 1.10 if and only if satisfies Equation 1.10. Moreover, one can easily check that

and

This means that corresponds to the duality between potentials and stream functions of Chaplygin gas flows such that

,

the density is given as Equation 1.18, and

for some constant .

When (resp., ), this gives a correspondence between graphs of minimal surfaces in and graphs of maximal surfaces in (resp., an involution on the set of graphs of time-like ZMC-surfaces in ) which we call the fluid mechanical duality.

A part of the above dualities is suggested in the classical book Reference 4. Calabi Reference 5 also recognized this duality for and pointed out the following:

Fact 1.1 (Calabi’s Bernstein-type theorem).

Suppose that the graph of a function gives a maximal surface (that is, a surface consisting only of space-like points whose mean curvature function vanishes identically). Then is linear.

This is an analogue of the classical Bernstein theorem for minimal surfaces in . Moreover, Calabi Reference 5 obtained the same conclusion for entire space-like ZMC-graphs in (), and Cheng and Yau Reference 6 extended this result for complete maximal hypersurfaces in for . The assumption that the graph consists only of space-like points is crucial. Entire ZMC-graphs which are not planar actually exist. Typical such examples are of the form

where is any -function of one variable. A point is a light-like point of if and only if . Moreover, if the graph of does not contain any light-like points, the potential function corresponding to is given by

up to a constant, where the sign coincides with that of . On the other hand, Osamu Kobayashi Reference 18 pointed out the existence of entire graphs of ZMC-surfaces with space-like points, light-like points, and time-like points all appearing. Such a surface is called of mixed type. Recently, many such examples were constructed in Reference 9.

By definition, any entire ZMC-graph of mixed type has at least one light-like point. So we give the following definition.

Definition 1.2.

A light-like point of the function (i.e., ) is said to be non-degenerate (resp., degenerate) if does not vanish (resp., vanishes) at .

At each non-degenerate light-like point, the graph of changes its causal type from space-like to time-like. This case is now well understood. In fact, under the assumption that the surface is real analytic, it can be reconstructed from a real analytic null regular curve in (cf. Gu Reference 12 and also Reference 11Reference 16Reference 17).

On the other hand, there are several examples of ZMC-surfaces with degenerate light-like points (cf. Reference 1Reference 2Reference 10Reference 14). Moreover, a local general existence theorem for maximal surfaces with degenerate light-like points is given in Reference 21. For such degenerate light-like points, we need a new approach to analyze the behavior of and . The following fact was proved by Klyachin Reference 17 (see also Reference 21).

Fact 1.3 (The line theorem for ZMC-surfaces).

Let be a domain of and let be a -differentiable ZMC-immersion such that is a degenerate light-like point. Then, there exists a light-like line segment passing through of such that does not coincide with one of the two end points of and contains , where is the set of degenerate light-like points of .

Recently, Fact 1.3 was generalized to a much wider class of surfaces, including constant mean curvature surfaces in ; see Reference 21Reference 22. (In Reference 21, the general local existence theorem of surfaces which changes their causal types along degenerate light-like lines was also shown.) The asymptotic behavior of along the line consisting of degenerate light-like points is discussed in Reference 21.

The purpose of this paper is to prove the following assertion:

Theorem A.

An entire -differentiable ZMC-graph which is not a plane admits a non-degenerate light-like point if its space-like part is non-empty.

This assertion is proved in Section 2 using the fluid mechanical duality and the half-space theorem for minimal surfaces in given by Hoffman-Meeks Reference 15. It should be remarked that the half-space theorem does not hold for time-like ZMC-surfaces. In fact, the graph of gives a properly embedded time-like ZMC-surface lying between two parallel vertical planes. In Section 2, we give further examples and provide a few questions related to Theorem A. As an application, we give the following improvement of Calabi’s Bernstein-type theorem:

Corollary B.

An entire -differentiable ZMC-graph which does not admit any time-like points is a plane.

In fact, if the ZMC-graph admits a space-like point, then the assertion immediately follows from Theorem A. So it remains to show the case that the graph consists only of light-like points. However, such a graph must be a plane, as shown in the appendix (see Theorem A.1).

2. Proof of Theorem A

In this section, we prove Theorem A in the introduction. We let be a -function satisfying the ZMC-equation Equation 1.10. We assume admits a space-like point but admits no non-degenerate light-like points. By Calabi’s Bernstein-type theorem (cf. Fact 1.1), has at least one degenerate light-like point. We set

which gives the ZMC-graph of . We denote by the positive semi-definite metric which is the pull-back of the canonical Lorentzian metric of by . The line theorem (cf. Fact 1.3) yields that the image of contains a light-like line segment . Then the projection of is a line segment on the -plane . Then lies on a line on . If , then there exists an end point of on . Since is the limit point of degenerate light-like points, itself is also a degenerate light-like point. By applying the line theorem again, there exists a light-like line segment containing as its interior point. We denote by the projection of to the -plane. Since the null direction at with respect to the metric is uniquely determined, also lies on the line . Thus, the entire graph contains a whole light-like line containing . In particular, degenerate light-like points on the graph consist of a family of straight lines in .

Let and be two such straight lines. Then never meets . In fact, if not, then there is a unique intersection point . By Fact 1.3, two lines can be lifted to two light-like lines and in passing through . The tangential directions of and are linearly independent light-like vectors at . Then by Reference 19, Lemma 27 in Section 5, is a time-like point, a contradiction.

Thus, the set of degenerate light-like points of consists of a family of parallel lines in the -plane. Without loss of generality, we may assume that these lines are vertical and one of them is the -axis. Then we can find a domain ()

such that and has no light-like points on and both of the lines and consist of light-like points unless . Since there are no light-like points on , the potential function is induced by as the fluid mechanical dual. The graph of is a minimal surface in . In particular, is real analytic. If we succeed in proving that the map is proper, then Theorem A follows. In fact, by the half-space theorem given in Reference 15 the image lies in a plane in . Then the map also lies in a plane in on . Since is light-like, the plane must be light-like, contradicting the fact that .

To prove the properness of , it is sufficient to show the following:

Lemma 2.1.

Let be a sequence of points in accumulating to a point on or . Then diverges.

Proof.

By switching the roles of and if necessary, it is sufficient to consider the case that accumulates to a point on . Taking a subsequence and using a suitable translation of the -plane, we may assume that converges to the origin and satisfies the following properties:

there exists such that for each and

there exists () such that

Since consists of degenerate light-like points, there exists a neighborhood of such that (see Reference 10 or Reference 21, (6.1))

where is a -differentiable function defined on (see Reference 21, Appendix A). Taking to be sufficiently small, we may assume that

Since , the potential function associated to satisfies (cf. Equation 1.18)

Since

is non-negative on the closure