ۜۨۨۤﮤﮡﮡ۞ۣ۩ۦۢٷ۠ۧﮠۗٷۡۖۦۘۛۙﮠۣۦۛﮡﯣﯞﯣ
ﯠۣۘۘۨۢٷ۠ ۧۙۦ۪ۗۙۧ ۣۚۦ
ẺỀẽẹẬặ Ẻằ ếẳẰ ẹẾếẴếỀếẰ Ẻằ ẬếẳẰẸẬếẴẮẾ Ẻằ ỀẾẾẴẰỀ
ﯗۡٷ۠ ٷ۠ۙۦۨۧﮤ ﯙ۠ۗﭞ ۜۙۦۙ ۑ۩ۖۧۗۦۣۤۨۢۧﮤ ﯙ۠ۗﭞ ۜۙۦۙ ﯙۣۡۡۙۦۗٷ۠ ۦۙۤۦۢۨۧﮤ ﯙ۠ۗﭞ ۜۙۦۙ ےۙۦۡۧ ۣۚ ۩ۧۙ ﮤ ﯙ۠ۗﭞ ۜۙۦۙ
ےﯜﯗ ﯙﯜېﯢۑےۍﯘﯘﯗﮐ ێېۍﯡﮐﯗﯞ ﯢﯟ ﮐۍېﯗﯟےﯧﯢﯠﯟ ﯛﯗۍﯞﯗےېﯦ
ﮐ۪ۙ ﮐۣۤۙۧ ۘۙ ﮐۡٷ ٷۢۘ ﯣۣۦۛۙ ﯜۙۦۖۙۦۨ ۑۣٷۦۙۧ ۘۙ ﮐۦٷ
ﯣۣ۩ۦۢٷ۠ ۣۚ ۨۜۙ ﯢۢۧۨۨ۩ۨۙ ۣۚ ﯞٷۨۜۙۡٷۨۗۧ ۣۚ ﯣ۩ۧۧۙ۩ ﮡ Џۣ۠۩ۡۙ Ң ﮡ ﯢۧۧ۩ۙ ڼڽ ﮡ ﯣٷۢ۩ٷۦۺ ھڼڼңﮞ ۤۤ үڽ ҰҰ ﯚۍﯢﮤ ڽڼﮠڽڼڽҮﮡۑڽۀҮۀҮۀүڼڼҢڼڼڼھڼۀﮞ ێ۩ۖ۠ۧۜۙۘ ۣۢ۠ۢۙﮤ ھڽ ۍۣۗۨۖۙۦ ھڼڼҢ
ﮐۢﭞ ۣۨ ۨۜۧ ٷۦۨۗ۠ۙﮤ ۜۨۨۤﮤﮡﮡ۞ۣ۩ۦۢٷ۠ۧﮠۗٷۡۖۦۘۛۙﮠۣۦۛﮡٷۖۧۨۦٷۗۨﮰۑڽۀҮۀҮۀүڼڼҢڼڼڼھڼۀ ﯜۣ۫ ۣۨ ۗۨۙ ۨۜۧ ٷۦۨۗ۠ۙﮤ
ﮐ۪ۙ ﮐۣۤۙۧ ۘۙ ﮐۡٷ ٷۢۘ ﯣۣۦۛۙ ﯜۙۦۖۙۦۨ ۑۣٷۦۙۧ ۘۙ ﮐۦٷ ڿھڼڼң۶ﮠ ےﯜﯗ ﯙﯜېﯢۑےۍﯘﯘﯗﮐ ێېۍﯡﮐﯗﯞ ﯢﯟ ﮐۍېﯗﯟےﯧﯢﯠﯟ ﯛﯗۍﯞﯗےېﯦﮠ ﯣۣ۩ۦۢٷ۠ ۣۚ ۨۜۙ ﯢۢۧۨۨ۩ۨۙ ۣۚ ﯞٷۨۜۙۡٷۨۗۧ ۣۚ ﯣ۩ۧۧۙ۩ﮞҢﮞ ۤۤ үڽҰҰ ۣۘﮤڽڼﮠڽڼڽҮﮡۑڽۀҮۀҮۀүڼڼҢڼڼڼھڼۀ
ېۙۥ۩ۙۧۨ ێۙۦۣۡۧۧۢۧ ﮤ ﯙ۠ۗﭞ ۜۙۦۙ
doi:10.1017/S1474748005000204 Printed in the United Kingdom
THE CHRISTOFFEL PROBLEM IN LORENTZIAN GEOMETRY
LEVI LOPES DE LIMA AND JORGE HERBERT SOARES DE LIRA
Departamento de Matem´atica, Universidade Federal do Cear´a, R. Humberto Monte, s/n, 60455-760, Fortaleza/CE, Brazil(levi@mat.ufc.br; jherbert@mat.ufc.br)
(Received 28 January 2004; accepted 26 October 2004)
Abstract The Christoffel problem, in its classical formulation, asks for a characterization of real func-tions defined on the unit sphereSn−1 ⊂Rn which occur as the mean curvature radius, expressed in
terms of the Gauss unit normal, of a closed convex hypersurface, i.e. the boundary of a convex body inRn. In this work we consider the related problem in Lorentz spaceLnand present necessary and
suffi-cient conditions for aC1function defined in the hyperbolic spaceHn−1⊂Lnto be the mean curvature
radius of a spacelike embeddingM֒→Ln.
Keywords:Christoffel problem; convex bodies; mean curvature radius; Lorentzian geometry
AMS 2000Mathematics subject classification:Primary 53C42; 53C50; 53A20
1. Introduction and statement of results
The Christoffel problem, in its classical formulation [3], asks for a characterization of real functionsφdefined on the unit Euclidean sphereSn−1⊂
Rnfor which the following occurs: there exists a bounded convex bodyK⊂Rnwhose boundaryM =∂Kis a closed
embedded hypersurface with the property that its mean curvature radius, expressed as a function of the Gauss unit normal, coincides withφpointwisely. After the contributions of many geometers (see [4] for some aspects of the history of the subject) the problem was at last solved in a satisfactory manner by Firey [3] and Berg [1]. The purpose of this paper is to discuss the Lorentzian analogue of this problem.
Recall that theLorentz spaceLn is the usualn-dimensional affine space with coordin-ates (x1, . . . , xn), endowed with the pseudo-Riemannian line element ds2= dx21+· · ·+
dx2
n−1−dx2n and corresponding metric·,·. A fundamental object in Lorentzian geome-try is thenull cone C={x∈Ln; x, x= 0}, and vectors inLn can in fact be classified by their position relatively toC: an elementx∈Ln is said to be spacelike(respectively, lightlike,timelike) ifx, x>0 (respectively, x, x= 0,x, x<0). The set of timelike vectors withxn >0 will be denoted byI+. Similarly, we setC+={x∈C; xn0}and
J+=I+∪C+.
Now assume that one has an isometrically embedded hypersurfaceMn−1inLn which
isspacelike in the sense that any tangent vector toM, after parallel translation to the
timelike vector, say ξ(p), determined by the conditions that ξ(p) is normal to TpM, i.e. ξ(p), v = 0 for any v ∈ TpM, and (again after parallel translation to the origin)
ξ(p) lies in the quadric Hn−1 ={x∈ Ln−1; x, x =−1, x
n >0}. This is the Gauss map ξ:M →Hn−1 of the embeddingM ֒→Ln.
In what follows we shall be interested in the case thatξis a diffeomorphism overHn−1.
This means that the embedding can be described by the inverse mapX =ξ−1:Hn−1→ M ֒→Ln. We call X thenormal representation of the embedding M ֒→ Ln. We also
impose the condition that, similarly to what happens in the classical formulation of the problem [3], M is the boundary of a convex body inLn. In order to make this point
precise, we introduce some more notation.
For each p ∈ Ln, let Lnp be the tangent space to Ln at p and τp : Ln0 → Lnp be the parallel displacement identifying Lnp to Ln0 = Ln. The null cone at p is given by
C(p) =τp(C) so that timelike, lightlike and spacelike vectors based atpare well defined. Similarly, we define C+(p), I+(p) and J+(p) and finally, given p = q in Ln, we set
pq={(1−t)p+tq; 0t1}andpq−→={p+tq; t0}(in the definition of−→pqwe may assumepnqn).
We now introduce a notion of convexity inLn which is well suited for applications to the corresponding Christoffel problem.
Definition 1.1. A set K ⊂ Ln is said to be L-convex (i.e. convex in the Lorentzian sense) if, givenp, q∈K (which we may assume to satisfy pnqn), one has
(i) ifq∈J+(p), then−→pq⊂K;
(ii) ifq /∈J+(p), thenpq⊂K.
Moreover, if intK=∅, we say thatK is anL-convex body.
Remark 1.2. Clearly, anL-convex set K⊂Ln is convex in the usual Euclidean sense. A typical example of anL-convex body isK={x∈Ln; x, x1, xn>0}. Notice that
∂K, the boundary of K, is preciselyHn−1. In fact, our definition is modelled after this
example. But anotherL-convex body one should have in mind isK={x∈Ln; x, x
0, xn 0}, whose boundary isC+.
We can now give a formulation of the Christoffel problem in Lorentzian geometry.
To characterize those functionsφ:Hn−1→Rfor which the following holds: there exists
an L-convex body K ⊂ Ln whose boundary M = ∂K ⊂ Ln is a spacelike embedding
described by a normal representation mapX :Hn−1→M as above, and such thatφ(u)
is the mean curvature radius of the embedding atX(u)for eachu∈Hn−1.
For convenience, a functionφas above will be termed a Christoffel function.
Even though a complete characterization of Christoffel functions seems to be a some-what involved problem, we present in this paper some partial results in this direction. In order to describe our main result, we recall thatHn−1, with the induced metric from
the embeddingHn−1֒→Ln, is a Riemannian manifold. In fact,Hn−1is the well-known
a distancedand for eachρ >0 we may consider the (n−2)-dimensional geodesic sphere
Sρ(u′) ={u∈Hn−1; d(u′, u) =ρ} centred at some u′ ∈Hn−1. We denote by dAρ the corresponding area element inSρ(u′) and set
A(ρ) =
Sρ(u
′)
dAρ=ωn−2sinhn−2ρ, (1.1)
the (n−2)-areaofSρ. Here,ωn−2is the area of the usual Euclidean sphereSn−2⊂Rn−1.
Moreover, for a continuous functionφ:Hn−1→Rwe define
Mρ(φ;u′) = 1
A(ρ)
Sρ(u
′)
φdAρ, (1.2)
themean value ofφoverSρ(u′). Following [6], we also pick the function
γ(ρ) = coshρ
ωn−2 ρ
+∞
dη
sinhn−2ηcosh2η, (1.3)
and introduce the kernel functionΓ :Hn−1×Hn−1→R∪ {∞}given by
Γ(u′, u) =γ(ρ), ρ=d(u′, u). (1.4)
We also recall that the usual action ofR+by dilations on the open future coneC+−{0}is
such that the corresponding orbit space is naturally identified to theboundary at infinity
∂∞Hn−1ofHn−1. Finally, letCbe the collection ofC2functionsφ:Hn−1→Rsatisfying
the following assumptions.
Assumption A. There exist constants Cj = Cj(φ) > 0, j = 1,2, such that for any
u′ ∈Hn−1 the bounds|M
ρ(φ;u′)|C1and|M˙ρ(φ;u′)|C2hold for ρ >0.
Assumption B. There exists C3=C3(φ)>0 such that, for anyρ >0, there holds
lim u′→∂
∞Hn−1
Mρ(φ;u′)C3, (1.5)
uniformly inu′.
Here and everywhere, the dot will stand for derivation with respect toρ.
We shall refer to the conditions above as Assumptions A and B, and the motivations behind their choice are scattered throughout the text (see Remarks 3.3 and 5.6 below). With this notation, we have the following theorem.
Theorem 1.3. A function φ ∈ C is a Christoffel function if and only if the following holds: for anyu′∈Hn−1, the quadratic form
Qu′(u′′) =
Hn−1
−γ− γ˙ sinhρu
′, u
φ(u) du
u′′, u′′
+
Hn−1 d dρ
γ˙
sinhρ
1
sinhρu
′′, u2φ(u) du, (1.6)
defined for u′′ ∈ T
u′Hn−1, satisfiesQu′(u′′) 0 and moreover there exists u′
0 ∈Hn−1
such thatQu′
It is not hard to check that our result covers the case in whichφis radially symmetric, and of course meets the conditions at infinity posed by Assumptions A and B. In this regard, an earlier contribution by Sovertkov [6] furnishes sufficient conditions in the case that φ is non-negative and constant outside a compact set. One should mention, however, that Sovertkov’s analysis only gives generalized solutions and thus overlooks features which are inherent to the problem (see the comments following Remark 2.2). On the other hand, geometric regularity issues are effectively treated here and thus in a sense our work should be considered as complementary to the analysis in [6].
Sovertkov also considers the uniqueness issue in Christoffel problem. In the classical situation it is known that if two compact convex bodies in Rn have the same mean curvature radius (as a function of the Gauss unit normal of their boundaries) then they differ by a translation [3]. This follows, for example, from the standard decomposition ofL2(Sn−1) in terms of spherical harmonics. A similar reasoning does not apply in the
Lorentzian setting and in factL2 methods do not seem to be adequate for the analysis
of the Christoffel problem inLn. For example, the standard embeddingHn−1֒→Ln has constant mean curvature radius given byr=n−1 (see Remark 3.2) and this is clearly not inL2(Hn−1). Even though Sovertkov displays a uniqueness criterion whenn= 3 and
in the generalized framework, the above comments should give some indication that the uniqueness question is very involved indeed.
Finally, we point out that the boundary of anL-convex body given by Theorem 1.3 is in fact a graph over the hyperplane {xn = 0}, and describing these boundaries non-parametrically implies that our Christoffel problem may be formulated as one of finding solutions to a fully nonlinear PDE with suitable asymptotic boundary conditions. It is then remarkable that our strategy, which involves justlinear equations, may in the end provide solutions to this nonlinear problem without appealing to the harder techniques ofa priori estimates and degree theory. We will present the non-parametric formulation of the Christoffel problem in Remark 2.3.
This paper is organized as follows. In§2 we introduce our basic set-up and derive the fundamental equations relating the mean curvature radius functionr to the geometry of a spacelike embeddingM ֒→Ln by means of certain elliptic equations. In §3 these
equations are solved, after replacingrbyφ∈ C, in a rather explicit manner, namely, in terms of convolution against suitable Green functions. In§4 we study a class of subsets inLn, theL-convex bodies, which in our work plays a similar role to the one by bounded convex bodies in the classical problem. Finally, in §5, we use the material previously developed in order to prove Theorem 1.3 by following the general lines of [3].
2. Spacelike hypersurfaces in Ln
and the normal representation
The purpose of this section is to derive the basic equations in our approach to the Christoffel problem, so we start by considering a spacelike hypersurfaceM ֒→Ln with
the property that the Gauss mapξ:M →Hn−1 is a diffeomorphism over Hn−1. This
allows us to regard the geometric data associated with the embedding as depending on the parameter u ∈ Hn−1. For example, as remarked in §1, the embedding itself can
representationof the embedding. Also of relevance is thesupport functionh:Hn−1→R
given by
h(u) =X(u), u, u∈Hn−1. (2.1)
In what follows, we will denote the covariant derivative of tensors in Ln by ∇. In particular,∇f means the gradient of a functionf defined in some open set inLn. It is useful to recall that
∇f = (∂1f, . . . ,−∂nf).
We still denote by ∆f the Laplacian inLn, i.e. ∆f = tr∇2f, where∇2f =∇∇f is the
Hessian off. The gradient, Hessian and Laplacian for functionsf defined inHn−1 will
be denoted, respectively, by∇0,∇20and ∆0.
In order to further explore the basic properties of these concepts, let us introduce appropriate coordinates inI+ ⊂Ln. For eachx∈I+, consider its radial projection given
by u = x/r ∈ Hn−1, where r = |x, x|1/2. Now, given u, u′ ∈ Hn−1, the hyperbolic
distance ρ=d(u′, u) betweenuandu′ is given byu= coshρu′+ sinhρω, for someω ∈ Sn−2(u′) ={x∈Ln : x, u′= 0, x, x= 1}. We then introduce radial coordinates in I+ centred atu′= (0, . . . ,0,1) by the correspondencex= (r, ρ, ω). In these coordinates,
the Lorentzian Laplacian ∆ inI+ reads as
∆ =−∂rr−
(n−1)
r ∂r+
1
r2∂ρρ+
A′(ρ) A(ρ)∂ρ+
1
r2sinh2ρ∆Sn−2, (2.2)
where ∆Sn−2 is the usual Laplacian in Sn−2. From this we infer that the Laplacian ∆0 inHn−1 can be expressed in radial coordinates centred atu′ as
∆0=∂ρρ+
A′(ρ) A(ρ)∂ρ+
1 sinh2ρ∆Sn
−2. (2.3)
Naturally, an identical decomposition for ∆0 holds in radial coordinates centred at any
other point ofHn−1.
For later reference, we also bring about the fact that, for ap-homogeneous function
ψ:I+→R(i.e. satisfyingψ(ru) =rpψ(u),r >0,u∈Hn−1), we have
∆ψ(u) =−p(n+p−2)ψ∗+ ∆0ψ∗, u∈Hn−1, (2.4)
whereψ∗ is the restriction ofψ toHn−1.
We now present a computation leading to a necessary condition for a function φ :
Hn−1 → R to be a Christoffel function. This is essentially an adapted version of the
corresponding calculation in the Euclidean case [2]. First, one has to extendhtoI+ as
a homogeneous function of degree one, so that it will make sense to speak about the partial derivatives∂h/∂xi (in what follows, we still denote the extended function by the same symbol). Similarly, X is extended to I+ as a 0-homogeneous Ln-valued function. Thus forv tangent toHn−1 it follows from (2.1) that
and since
∇vX, u= 0 (2.5)
becauseuis normal to the embedding atX(u), one has
∇0h, v=∇h, v=X,∇vu, (2.6)
or, more succinctly, omittingv,∇0h=X,∇u. Differentiating again with respect to a
tangent direction, we have, after omittingv, that
∇2
0h=∇X,∇u+X,∇2u.
Using that∇2u=uId, one has
∇20h−hId =A, (2.7)
where Id : TuHn−1 → TuHn−1 is the identity map and A = ∇X,∇u is the second fundamental form of the embedding. If we diagonalize Awith respect to the standard Riemannian metric inHn−1, one has
A=
⎛
⎜ ⎝
r1 . . . 0 ..
. . .. ... 0 . . . rn−1
⎞
⎟ ⎠,
whereri is the ith curvature radius of the embeddingX. If we take traces in (2.7) and set
r=r1+· · ·+rn−1,
themean curvature radius of the embedding X, we have
∆0h(u)−(n−1)h(u) =r(u), u∈Hn−1,
and the result of the computation can be recorded as follows.
Proposition 2.1. IfX :Hn−1→Ln is the normal representation of a spacelike
embed-dingM ֒→Lnas above andr:Hn−1→Ris the mean curvature radius of the embedding, one has, inHn−1,
∆0h−(n−1)h=r, (2.8)
or equivalently, inLn,
∆h=r, (2.9)
whererhas been extended as a(−1)-homogeneous function.
Remark 2.2. Notice that a crucial step in the above calculation is the identity (2.5), as this allows us to relaterandhvia the elliptic equation (2.8). Notice moreover that (2.5) essentially expresses the fact that the mapX :Hn−1→Ln is the normal representation of the embedding M =X(Hn−1) or equivalently that the unit Gauss normal to M at
Proposition 2.1 gives the basic differential equation in our approach to the Christoffel problem, namely, the relationship between the support functionhand the mean curvature radius r. Notice, moreover, that, in view of (2.1), since h is 1-homogeneous, we have by Euler’s formula that if h is given we can recover the normal representation of the embedding by
X =∇h, (2.10)
or equivalently,
X(u′) =−h(u′)u′+∇
0h(u′), u′∈Hn−1. (2.11)
This suggests an effective approach to the Christoffel problem: first one solves (2.8), with rreplaced by φ ∈ C, and then we define X by (2.10). But one should be aware of the intrinsic difficulties inherent to this approach, the basic observation being that it is not at all clear from (2.10) thatX defines a regular embedding ofHn−1 inLn in the sense that it has maximal rank everywhere and moreover thatX is the normal representation of its imageX(Hn−1) (see Remark 2.2). In this work we deal with this problem by adapting
Firey’s argument to our setting.
Notice that it follows from (2.9) and (2.10) that
∆X = ∆∇h=∇∆h=∇r,
so we also may formulate the Christoffel problem in terms of a Poisson type equation ∆X =∇r. Firey’s approach in [3] consists precisely of finding an explicit Green function for this problem. However, a direct adaptation of this approach to our case does not seem to work so we deal instead with (2.8) directly.
Remark 2.3. In §4, we will prove that the boundary of an L-convex body K (see Definition 1.1) is a complete graph over the hyperplane Σ = {xn = 0}. Thus, if we describe the boundary ∂K of this body as a graph of a functionxn = f(x1, . . . , xn−1)
defined in a domain ofΣ, we obtain that the fundamental forms of∂K are given by
I(∂i, ∂j) =δij− ∇if∇jf,
II(∂i, ∂j) = ∇i∇jf
W2 ,
III(∂i, ∂j) =W2II2(∂i, ∂j),
whereW =
1− ∇f,∇fand{∂i}are the coordinate fields. This means that the basic equation (2.8) in terms off becomes
i
j=ifjj
jfjj
= (n−1)r. (2.12)
We remark that this fully nonlinear equation corresponds to the curvature equation
3. Solving the equation for the support function
In this section we start the proof of Theorem 1.3 by solving equation (2.8), withrreplaced byφ∈ C, for the support function of a spacelike embedding whose mean curvature radius function, as we show in§5, is preciselyφ.
The problem is that of finding an explicit solution for the elliptic equation
Lh=φ, (3.1)
for a givenφ∈ C, whereL= ∆0−(n−1). Now, (2.3) is of some help in finding radial
solutionsγ=γ(ρ) of the homogeneous equationLγ=δu′ since this reduces to
¨
γ+A˙
Aγ˙ −(n−1)γ= 0. (3.2)
As remarked in [6], a solution for this ordinary differential equation is given by (1.3). For the sake of completeness, we include the argument in [6] showing that, in the case when
φ∈ C,h:Hn−1→Rgiven by
h(u′) =
Hn−1
Γ(u′, u)φ(u) du (3.3)
is a solution to (2.8). By standard elliptic theory, one just has to prove thathgiven by (3.3) is a formal solution of (2.8). Using radial coordinates centred atu′, (3.3) can be
rewritten as
h(u′) =
+∞
0
Sρ(u
′)
Γ(u′, u)φ(u) dµ(u),
where dµ(u) = sinhn−2ρdω is the area element of Sρ(u′) (here, dω is the standard area element ofSn−2⊂Rn−1). From (1.4) we have
h(u′) =
+∞
0
dρ
Sρ(u
′ )
γ(ρ)φ(u) dµ(u)
=
+∞
0
γ(ρ) dρ
Sρ(u
′)
φ(u) dµ(u)
=
+∞
0
γ(ρ)A(ρ)Mρ(φ;u′) dρ,
so that differentiating under the integral sign and using (2.3) we obtain
Lh(u′) =
+∞
0
γ(ρ)A(ρ)L(Mρ(φ;u′)) dρ
=
+∞
0
γ(ρ)A(ρ)
¨
Mρ(φ;u′) + ˙
A(ρ)
A(ρ)M˙ρ(φ;u
′)−(n−1)M
ρ(φ;u′)
We now dropu′ andφfrom our notation, so that
Lh(u′) =
+∞
0
γAM¨ρdρ+
+∞
0
γA˙M˙ρdρ−(n−1)
+∞
0
γAMρdρ
=γAM˙ρ|+0∞− +∞
0
˙
γAM˙ρdρ−(n−1)
+∞
0
γAMρdρ
=γAM˙ρ|+∞
0 −γAM˙ ρ|+0∞+ +∞
0
(¨γA+ ˙γA˙−(n−1)γA)Mρdρ
=γAM˙ρ|+0∞−γAM˙ ρ|+0∞,
where we have used (3.2) and integration by parts twice. Now, from (1.1) one easily checks by l’Hˆopital’s rule that
lim
ρ→0+γA= limρ→+∞γA= limρ→+∞γA˙ = 0
and
lim
ρ→0+γA˙ = 1,
so that from Assumption A we finally get
Lh(u′) =M
ρ=0(φ;u′) =φ(u′),
and we can record the result of our computation as follows.
Proposition 3.1. Ifφ∈ C, then the functionh:Hn−1→Rgiven by (3.3) is a solution
of (3.1). Moreover, one has
|h(u)| C1
n−1, u∈H
n−1, (3.4)
whereC1is the constant appearing in Assumption A.
Proof . The only point that remains to be checked is the uniform boundedness ofh. For this first notice that one has
h(·) =
+∞
0
˜
γ(ρ)Mρ(φ,·) dρ, (3.5)
where ˜γ(ρ) =γ(ρ)A(ρ). Now, (1.3) implies that ˜γ0 and, moreover, one has
+∞
0
˜
γ(ρ) dρ=− 1
n−1, (3.6)
and the result follows immediately.
Remark 3.2. Here is the rationale behind (3.6). If one looks at the standard embedding
Remark 3.3. Notice that the control onφ provided by Assumption A is crucial here, as it not only allows us to have an explicit solution for (3.1) but also because it gives the uniform bound (3.4), an essential point in what follows.
4. L-convex bodies in Lorentz space
Our aim in this section is to explore the notion ofL-convexity described in Definition 1.1, as it seems to be well suited for the applications to the Lorentzian Christoffel problem formulated in§1.
Proposition 4.1. IfK⊂Ln is anL-convex body, then
(1) ifp∈intK, thenJ+(p)⊂K;
(2) ifp∈∂K, thenI+(p)⊂K.
Proof . (1) Since p ∈ intK, there exists a small ball B ⊂ K centred at p. For each
q∈B∩J+(p),q= p, one has byL-convexity−→pq⊂K. But clearly
J+(p) =
q∈B∩J+(p)
− →
pq.
(2) Takeq ∈intK. If p∈I+(q), observe that I+(p)⊂I+(q)⊂J+(p)⊂K by (1), so
we are done. If p /∈ I+(q), notice that for each r ∈ I+(p)∩I+(q) one has −→pr⊂K by
L-convexity and, moreover, that
I+(p) =
r∈I+(p)∩I+(p)
− →
pr.
A basic principle in the classical structure theory of convex sets [2] is that abounded convex body is the envelope of its support hyperplanes, and one would like to determine natural conditions in order to have a similar result here, regardless of the fact thatL -convex bodies are necessarily unbounded. In the Lorentzian case, however, one should be careful enough in order to distinguish the possibilities that arise for the candidates to support hyperplanes.
Definition 4.2. An affine hyperplane Π ⊂ Lnp passing through p is spacelike if Π ∩
C(p) ={p}. In other words, there exists a non-zero timelike vector u ∈ Lnp such that
Π ={p+tv; u, v= 0}. In case Π∩C(p) is a line of lightlike vectors (a generator of
C(p)) we say that Π is degenerate. Finally, if any of these possibilities occurs, we say thatΠ isachronal.
Remark 4.3. Note that an achronal hyperplane Π splits Ln into two parts so that it makes sense to say when a setS ⊂Ln liesabove Π. By this we mean of course that S
lies entirely in that part ofLn containingJ+(p) for somep∈Π. Similarly, we can also
Definition 4.4. GivenS⊂Lnandp∈∂S, an achronal hyperplaneΠ ⊂Lnp is asupport hyperplane forS ifS lies entirely aboveΠ.
Proposition 4.5. If K ⊂ Ln is an L-convex set, there exists at least one support hyperplane passing through eachp∈∂K.
Proof . The proof is by induction innso we examine first the case n= 2. LetK ⊂L2 be an L-convex body and consider p∈ ∂K. By Proposition 4.1, take q ∈I+(p) ⊂ K,
q=p, and consider the linelof timelike vectors determined bypandq. Now, pdivides
l into two open half-lines, say l+ and l−, and assume that q∈ l+ so that l± ⊂I±(p).
The idea now is to consider two copies ofl−, sayl−
1 andl−2, and move them, l1− to the
left andl−
2 to the right, while keepingpfixed, in the future direction until that a point
inK (actually in ∂K) is reached. We claim that the limiting positions ofl−
1 and l−2 are
such that the angle between them is at leastπ. Otherwise, due to Proposition 4.1 (2), one would find pointsq1, q2 ∈intK, close tol1− and l−2, respectively, such thatpq1 and pq2
make an angle strictly less thanπ. Now encloseq1andq2in open small ballsB1, B2⊂K,
respectively, and letQbe the convex hull (in the usual Euclidean sense) ofB1,B2andp.
It is now easy to exhibitr∈Q⊂K lying inl−, a contradiction sincel−∩K =∅. The
claim is thus established and this gives us enough room to draw an achronal linel′ which
is a support line forKat p.
We now turn to the general case, so thatK ⊂Ln, n 3, is an L-convex body. As before, givenp∈∂K, takeq∈I+(p)⊂K,q=p, and consider any hyperplaneΠ passing
throughpandq. Since−→pqis formed by timelike vectors,Π is a copy ofLn−1. Moreover,
K∩Π is an L-convex body in Π (i.e. intK∩Π = ∅) and we can use the induction assumption toK∩Π so that there exists an (n−2)-dimensional achronal planeσ⊂Π
which supportsK∩Π at pinside Π. IfΠ− is the open half-plane inΠ determined by σand such that Π−∩K =∅, we may easily complete the proof by moving two copies
ofΠ− in the same way as described before for the half-linel− in the casen= 2, while
keepingσfixed.
Remark 4.6. Notice that in the proof above for the case n= 2, it might well happen thatl′ ⊂C(p). This is the case for example if one takesK=C+ andp∈C+,p= 0, and
this accounts for the necessity of including degenerate hyperplanes in Definition 4.4.
Remark 4.7. Even though it is not essential in what follows, we mention that it is possible to proceed in describing finer properties of L-convex bodies. For example, it holds that any such subset hasC0 boundary, but we shall not pursue this here.
5. Recovering the embedding
In this section we complete the proof of our main result by showing in particular that any φ : Hn−1 → R as in Theorem 1.3 arises as the mean curvature radius function
r of a suitable spacelike embedding M ⊂ Ln which is the boundary of an L-convex body inLn. A natural candidate for a parametrization of the embedding is the mapping
route we shall take, so we are left with two tasks. First, one has to check thatXso defined is a regular map. Second, one has to show that X is indeed the normal representation of its image X(Hn−1), as this will allow us to compute r in terms of h and to prove
that r=φidentically (see Remark 2.2), a crucial step in the final checking that r=φ identically. It is precisely here that the non-negativity hypothesis on the quadratic form Qgiven by (1.6) and Assumption B onφare needed.
In the light of classical constructions in the theory of convex bodies [2], it is natural to expressX(Hn−1) as the boundary of a certainL-convex body inLn manufactured out ofh. More precisely, we set
K=
u∈Hn−1
Π+(u), (5.1)
where
Π+(u) ={x∈Ln; x, uh(u)}. (5.2)
Notice that each Π+(u) is a closed half-space in Ln whose boundary is the spacelike hyperplane
Π(u) ={x∈Ln; x, u=h(u)} (5.3)
so thatKis anL-convex set ofLn indeed. Moreover, the timelike normal vector toΠ(u) is preciselyu.
At this point, however, a departure from the classical methods is needed, for in the classical settinguvaries in a compact space, the unit sphereSn−1⊂Rn, and this helps to show that∂Kis the envelope of its support hyperplanes (see [2]). In order to show the corresponding fact in our setting, namely, that asuvaries onHn−1, every Π(u) occurs
as a support hyperplane forK defined by (5.1) and that in fact these are the only ones, a more involved argument is required. We start by observing that (2.10) readily implies
X =∇h, (5.4)
thus yielding
∇X =∇2h, (5.5)
where
∇X =
⎛
⎜ ⎝
∇X1
.. . ∇Xn
⎞
⎟ ⎠=
⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
∂X1
∂u1 . . .
−∂X1
∂un ..
. . .. ...
∂Xn
∂u1
. . . −∂Xn
∂un
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(5.6)
is the Lorentzian Jacobian ofX and∇2his the Lorentzian Hessian ofh(see§2), where
of courseX and hhave been extended toI+ as homogeneous functions of degrees zero
Proposition 5.1. Ifφ∈ C, one has, foru′ ∈Hn−1,
∇h(u′) =−h(u′)u′−
Hn−1
˙
γ
sinhρ(u+u, u
′u′)φ(u)
du. (5.7)
Proof . In what follows, we denote the gradient ofΓ(u′, u) as a function ofu′ by∇
1Γ.
Forv∈Tu′Hn−1, (3.3) clearly gives
∇h(u′), v=
Hn−1
∇1Γ(u′, u), vφ(u) du
=
Hn−1 ˙
γ∇1ρ(u′, u), vφ(u) du.
If we write u = coshρu′ + sinhρω, for some ω ∈ Sn−2(u′), then we obtain coshρ =
−u, u′, and differentiating we get
∇1ρ(u′, u), v=−
v, u
sinhρ, (5.8)
so that
∇h(u′), v=−
Hn−1 ˙
γ
sinhρv, uφ(u) du. (5.9)
If{vi}n−1
i=1 is an orthonormal basis ofTu′Hn−1, we conclude that the tangential
compo-nent of∇h(u′) is
i
∇h(u′), viv
i=−
Hn−1 ˙
γ
sinhρ
i
vi, uvi
φ(u) du
=−
Hn−1 ˙
γ
sinhρ(u+u, u
′u′)φ(u) du.
Now, sincehis homogeneous of degree one, Euler’s relation gives
∇h(u′), u′=h(u′),
thus completing the proof.
It is clear from (5.5) that∇X =∇2his a symmetric matrix and we have the following
proposition.
Proposition 5.2. Ifφ∈ C, then the quadratic form associated with∇X is given byQ
in (1.6).
Proof . By Euler’s relation,∇2hvanishes along directions normal toHn−1, so it suffices
to compute∇2halong tangent directions. Forv, w∈T
u′Hn−1 we consider a unit speed
transport ofwinLn alongβ, we have
∇v∇h(u′), w=v∇h(u′), w − ∇h(u′),∇vw
=v∇h, w
= d
ds∇h(β(s)), W(s)|s=0
= d
ds∇h(β(s)), W(s) ⊤|s
=0+ d
ds∇h(β(s)), W(s) ⊥|s
=0
=I1+I2,
where W =W⊤+W⊥ is the decomposition into tangential and normal components.
From (5.7) we have
d
ds∇h(β(s)), W(s) ⊤=−
Hn−1 d ds
γ˙(ρ(β(s), u))
sinhρ(β(s), u)W(s)
⊤, u
φ(u) du
=−
Hn−1 d dρ
γ˙
sinhρ
∇1ρ(u′, u), vW(s)⊤, uφ(u) du
−
Hn−1 ˙
γ
sinhρ
d dsW
⊤(s), u
φ(u) du
=−I3(s)−I4(s),
so that by (5.8) we get
I3(0) =−
Hn−1 d dρ
γ˙
sinhρ
1
sinhρu, vw, uφ(u) du,
and expandingW⊤=W+W, ββ, we also get after a simple computation that
I4(0) =
Hn−1 ˙
γ
sinhρw, vu
′, uφ(u) du.
On the other hand,
d
ds∇h(β(s)), W(s) ⊥= d
ds(W(s), β(s)∇h(β(s)), β(s))
=−d
ds(W(s), β(s)h(β(s)))
=−W(s), β′(s)h(β(s))− W(s), β(s)d
dsh(β(s)),
and, sincew, u′= 0, we have
I2=−w, vh(u′).
Putting all the pieces of our computation together and taking (3.3) into account, the
The proof of the sufficiency condition in Theorem 1.3 now follows essentially the general lines in [3], with the necessary adjustments due to the Lorentzian nature of our metric. First, the non-negativity assumption on Qu′ as u′ varies over Hn−1 implies, in view
of (1.6) and Proposition 5.2, thath:I+→Risconvex in the sense that
h((1−λ)u+λv)(1−λ)h(u) +λh(v), u, v∈I+, λ∈[0,1].
This, together with the 1-homogeneity ofh, gives, in view of (5.4),
X(u′) =∇h(u′)∈Π(u′), u′∈Hn−1. (5.10)
In particular,K=∅. In fact, eachX(u′)∈∂K sinceK lies entirely aboveΠ(u′).
In order to proceed, one must rule out the presence of degenerate support hyperplanes forK, in conformity with the comments preceding (5.4).
Proposition 5.3. Every spacelike support hyperplane for K as in (5.1) is of the form
Π(u′), for someu′∈Hn−1.
Proof . In effect, if Π is a spacelike support hyperplane at p, let u′ ∈ Hn−1 be its
timelike unit normal vector. It is clear from the definition ofKand the comments above
thatΠ =Π(u′).
By Proposition 4.1, anyp ∈ ∂K admits a support hyperplane. We now use this to complement Proposition 5.3.
Proposition 5.4. Any support hyperplane forK as in (5.1) is spacelike.
The proof of this proposition, which uses the series of lemmas below, goes via contra-diction, so we start by assuming the existence of p∈∂K and Π a degenerate support hyperplane forK atp.
Lemma 5.5. If Π is a degenerate support hyperplane forK at some point p, then Π
contains the origin0∈Ln.
Proof . Given u′ = (u′
1, . . . , u′n)∈ Ln, set ˜u′ = (−u′1. . . ,−u′n−1, u′n). It follows that if
u′ is the unit timelike normal to Π(u′), then ˜u′/u˜′ is the unit normal, in the usual
Euclidean sense, toΠ(u′) (here, · is the norm coming from the usualEuclidean inner
product ‘·’ inRn =Ln.) By (5.10), the Euclidean equation forx∈Π(u′) is
(x− ∇h(u′))· u˜
′
u˜′ = 0,
or equivalently, in view of the 1-homogeneity ofh,
x· u˜
′
u˜′ =−
1 u˜′h(u
′). (5.11)
But for anyx∈Π(u′), the left-hand side above gives, up to a sign, the Euclidean distance d(0, Π(u′)) between the origin andΠ(u′) so that using (3.4) we get thatd(0, Π(u′))→0
asu′→∂
∞Hn−1. We now appeal to the bound (3.4) and apply (5.11) to a sequence{u′n} withΠ(u′
n) converging toΠ. SinceΠ is lightlike, it follows thatun′ →∂∞Hn−1 and we
Remark 5.6. The point of this lemma is that in the presence of the bound (3.4), which follows from Assumption A, degenerate support hyperplanes necessarily pass though the origin. To illustrate this, recall that from the explicit formulae in§3, ifφ=k, a positive constant, one has hk = −k/(n−1) for the corresponding support function and the associated embedding is given by Xk(u′) = ku′/(n−1), a homothetic copy of Hn−1 inside I+. Moreover, as k → 0, the sequence X
k converges towards the lightcone C+ and, outside a bounded region, their spacelike support hyperplanes approach degenerate ones (which are support hyperplanes toC+). But notice however thath
k→0, and this is precisely what is prevented by Assumption B.
Using the above lemma, we can consider the lightlike linellying inΠ, passing through 0 and parallel to the line generating the future light coneC+(p). Moreover, letl⊥ be the
Euclidean orthogonal complement to l in Π (so that l⊥ is spacelike in particular). We
now takeE0 andE1 to be mutually orthogonal vectors inLn satisfyingE0, E0=−1,
E1, E1= 1 and such thatLn, as a vector space, is generated byE0,E1 andl⊥. Ifu′is
an arbitrary vector in the intersectionΞ ofHn−1and the Lorentzian planeL2generated
byE0 and E1, and if σ is the arclength parameter for Ξ with σ= 0 corresponding to
E0, we haveu′= coshσE0+ sinhσE1. We now choose an adapted frame toTu′Hn−1 by
setting e0 =u′, e1 = sinhσE0+ coshσE1, the velocity vector at u′ of the unit speed
geodesic joiningE0 to u′, and{e3, . . . ,en−1} to be an orthonormal basis of l⊥ parallel
translated to u′. Now, if u′ ∈ Hn−1 ∩L2 as above, one has that Π(u′) is a support
hyperplane forK with unit normalu′ and intersectingΠ along a codimension one affine
subspace denoted byl⊥(u′) andparallel tol⊥. IfΠ+ is the closed half-space determined
byΠ and such thatK⊂Π+, we have
K⊂Π+∩Π+(u′). (5.12)
In particular,X(u′) =∇h(u′)∈Π+∩Π+(u′). Notice also that, by Proposition 4.1 (2),
the half-linel+{p} ∩C+(p) lies entirely inKand this means that, by replacingpbyτ p
for someτ >0 large enough, we may assumep, E0<0 andp, E1>0.
Lemma 5.7. Asσ→+∞, one has the uniform bounds
−∇h(u′), E0−p, E0, ∇h(u′), E1p, E1, (5.13)
Proof . Ifu′as above approaches∂
∞Hn−1(or equivalently, asσ→+∞), the coordinates
in the directions of E0 and E1 of points inl⊥(u′) increase in absolute value. But since
p∈Π andΠ(u′) is a support hyperplane forK,pshould remain abovel⊥(u′). And since
∇h(u′) ∈ Π(u′)∩Π+∩Π+(u′) and Π is a support hyperplane for K, ∇h(u′) should
remain belowl⊥(u). The result follows.
We will show in the sequel that under Assumptions A and B the bounds (5.13) are impossible, thus reaching a contradiction to our hypothesis on the existence of degenerate support planes. In terms of the adapted frame{ei}n−1
i=0 above, we have
∇h(u′) =−h(u′)u′+∇h(u′),e
1e1+
n−1
i=2
∇h(u′),eie
so that
∇h(u′), E0= coshσ(h(u′)−tanhσ∇h(u′),e1) (5.14)
and
∇h(u′), E
1= coshσ(−h(u′) tanhσ+∇h(u′),e1). (5.15)
We shall use these expressions to contradict the estimates (5.13).
Lemma 5.8. In the conditions of Theorem 1.3, we haveφ0.
Proof . By Proposition 5.2, the assumptions in Theorem 1.3 imply that ∇2h is
non-negative as a quadratic form. Thus, by (2.9),φ= ∆h0.
Lemma 5.9. Asσ→+∞,∇h(u′),e
1remains positive, bounded away from zero and
uniformly bounded from above.
Proof . From (3.5) we have
lim σ→+∞h(u
′) =
+∞
0
˜
γ(ρ) lim
σ→+∞(Mρ(φ;u
′))dρC′,
forC′ =−C
3/(n−1)<0. Here we have used (3.6) and Assumption B. Sinceu′ =e0,
this can be rewritten forσlarge enough as∇h(u′),e
0=h(u′)C′′ for someC′′<0.
But e0−e1 = (coshσ−sinhσ)(E0−E1) and this goes to zero as σ → +∞. Thus,
∇h(u′),e
1C′′′>0 in this range ofσ.
We now look at the upper bound for ∇h(u′),e
1. We first observe that a simple
computation starting from (1.3) gives
˙
γ=− 1
ωn−2
sinh2−nρ
coshρ + tanhργ,
where bothγ and ρ depend on the pair (u′, u). On the other hand, we can write u=
coshρu′+ sinhρw, wherew∈Sn−2(u′) is the initial speed of the geodesic joiningu′tou.
It follows thatu,e1= sinhρ(w,e1). Combining this with (5.9) and integrating radially
we get
|∇h(u′),e1| +∞ 0
tanhργ(ρ)A(ρ)Mρ(φ;u′)(w,e1) dρ + +∞ 0
sinh2−nρ
coshρ A(ρ)Mρ(φ;u ′)(w,e
1) dρ − +∞ 0
γ(ρ)A(ρ)Mρ(φ;u′) dρ+
+∞
0
1
coshρ|Mρ(φ;u ′)|dρ
−h(u′) +C
1 +∞ 0 dρ coshρ C1 1
n−1 +
π
2
where in the second inequality we have used ˜γ=γA0 and Lemma 5.8, which implies
Mρ(φ;u)0.
As a consequence of Lemma 5.9, one has the estimates
−∞< lim σ→+∞
h(u′)− sinhσ
coshσ∇h(u ′),e
1
M1
and
M2 lim
σ→+∞
−h(u′)sinhσ
coshσ+∇h(u ′),e
1
<+∞,
for constantsM1<0 andM2>0, so that using (5.14) and (5.15) we finally get
lim
σ→+∞∇h(u ′), E
0=−∞
and
lim
σ→+∞∇h(u ′), E
1= +∞,
thus contradicting (5.13) and completing the proof of Proposition 5.4.
Taken together, Propositions 5.3 and 5.4 say that only spacelike hyperplanes, which are always given by (5.3), are allowed as support hyperplanes forK and moreover that any such hyperplane so occurs. In particular, any element in∂K lies in someΠ(u) and thushis the support function ofK in the usual geometric sense [2], i.e.
h(u′) = max
x∈Kx, u
′, u′∈Hn−1, (5.16)
and moreover for eachu′ the maximum is attained at∂K.
We are now going to show thatK∩Π(u′) in fact reduces toX(u′). To this effect notice
that the expression ∇h(u′), v, as a function of v ∈I+, is the support function of the
point ∇h(u′) = X(u′) in the sense of (5.16). On the other hand, this same expression
gives the support function, again in the sense of (5.16), of the intersectionK∩Π(u′). The
conclusion is that, asu′ varies in Hn−1, K∩Π(u′) ={∇h(u′)} ={X(u′)} as desired,
and this shows that the map X : Hn−1 → ∂K is surjective and has the property that
the support hyperplane toKatX(u′) is preciselyΠ(u′) and hence hasu′ as its timelike
unit normal.
Now, the assumption on the existence ofu′
0∈Hn−1such thatQu′
0 >0 in Theorem 1.3 implies that intK = ∅, i.e. K is an L-convex body in Ln. In view of what we have proven in the last paragraph, this implies thatM =∂K =X(Hn−1)֒→Ln is a
space-like embedding whose normal representation is preciselyX (see Remark 2.2) and from Propositions 2.1 and 3.1 we conclude thatr=φidentically, as desired.
As for the necessity condition in Theorem 1.3, the proof is identical to the one in [3] and in fact it holds for anyC1functionφ:Hn−1→Rwhich can be realized as the mean
Acknowledgements.L.L.d.L. was partially supported by CNPq, FINEP and PRONEX. J.H.S.de L. was partially supported by FINEP and PRONEX.
The authors thank H. Rosenberg for helpful comments and suggestions and also for bringing Firey’s paper [3] to their attention. J.H.S.de L. thanks Professor Gerv´asio Colares for the encouragement dispensed to him during the last few years.
References
1. C. Berg, Corps convexes et potentials sph´eriques, Mat.-Fys. Medd. K. Dan. Vidensk. Selsk.37(1969), 1–64.
2. T. Bonnesen and W. Fenchel,Theorie der konvexen K¨orper (Springer, 1971). 3. W. J. Firey,The determination of convex bodies from their mean radius of curvature
functions,Mathematika 14(1967), 1–13.
4. W. J. Firey,Subsequent work on Christoffel’s problem about determining a surface from local measurements, in E. B. Christoffel: the influence of his work on mathematics and the physical sciences(ed. P. L. Butzer and R. F. Feher), pp. 721–723 (Birkh¨auser, Basel, 1981).
5. B. O’Neill,Semi-Riemannian geometry(Academic Press, 1983).
6. P. I. Sovertkov,The Christoffel problem in pseudo-Euclidean spaceEn−1,1,Math. Notes
