On fine differentiability properties of horizons and applications to Riemannian geometry

Piotr T. Chruściel^{∗} Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France

Joseph H. G. Fu^{†} Department of Mathematics University of Georgia Athens, GA 30602, USA

Gregory J. Galloway^{‡} Department of Mathematics University of Miami Coral Gables FL 33124, USA

Ralph Howard^{§} Department of Mathematics University of South Carolina Columbia S.C. 29208, USA

May 14, 2001

Abstract

We study fine differentiability properties of horizons. We show that the set of end points of generators of a n-dimensional horizon ℋ (which is included in a (n+1)-dimensional space-time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1 ≤ k ≤ n + 1 we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is “almost a C^{2} manifold of dimension n + 1 − k”: it can be covered, up to a set of vanishing (n + 1 − k)-dimensional Hausdorff measure, by a countable number of C^{2} manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.

∗Supported in part by KBN grant # 2 P03B 130 16. E–mail: Chrusciel@Univ-Tours.fr †Supported in part by NSF grant # DMS-9972094. E–mail: fu@math.uga.edu ‡Supported in part by NSF grant # DMS-9803566. E–mail: galloway@math.miami.edu §Supported in part by DoD Grant # N00014-97-1-0806. E–mail: howard@math.sc.edu

1 Introduction.

Horizons are amongst the most important objects that one encounters in causality theory: Cauchy horizons constitute boundaries beyond which predictability breaks down; event horizons are boundaries beyond which no return is possible. The key structural property of horizons is the existence of generators: recall that an embedded hypersurface ℋ ⊂ M is said to be future null geodesically ruled if every point p ∈ ℋ belongs to a future inextensible null geodesic Γ ⊂ ℋ; those geodesics are called generators of ℋ. One can then extract the essential properties of Cauchy horizons, or black hole event horizons, in the following definition: ℋ is a future horizon if ℋ is an achronal, closed, future null geodesically ruled topological hypersurface. It follows from the above definition (or from the properties of past Cauchy horizons, or from the properties of future event horizons) that the generators can have past endpoints on ℋ, but no future endpoints.

The set ℋ_{end} of end points of generators of ℋ provides an important tool in the study of the structure of horizons; for simplicity we will refer to those points as end points. In particular one wants to know how “large” this set can be. One defines the multiplicity N(p) of a point p ∈ ℋ as the number of generators which pass through or exit ℋ at p; it is well known that if N(p) > 1, then p is necessarily the endpoint of all relevant generators. The set of points with multiplicity N(p) > 1 determines the differentiability properties of horizons: as has been shown by Beem and Królak [2], horizons are non-differentiable precisely at this set. It is also well known that the set of points at which a horizon is non-differentiable has vanishing n-dimensional Hausdorff measure, and this gives one control over the size of the set of endpoints with multiplicity N(p) > 1. Thus, in order to control the dimension of ℋ_{end} it remains to estimate that of the set of endpoints with multiplicity N(p) = 1. Let us denote by ℋ_{diff} the set of points of ℋ at which ℋ is differentiable; what has been said shows that the set of endpoints with multiplicity N(p) = 1 coincides with the set ℋ_{end} ∩ ℋ_{diff}. Królak and Beem [2] have displayed an example of a horizon with an end point with multiplicity one (at which ℋ is, of course, differentiable), thus there exist horizons for which the set ℋ_{end} ∩ ℋ_{diff} is not empty. Our first main result is the following:

Theorem 1 Let ℋ be a future horizon in an (n + 1)-dimensional spacetime (M, g). Then the set ℋ_{end} ∩ ℋ_{diff} has vanishing n-dimensional Hausdorff measure. Moreover, for any C^{2} spacelike hypersurface 𝒮 the set ℋ_{end} ∩ ℋ_{diff} ∩ 𝒮 has vanishing (n − 1)-dimensional Hausdorff measure.

Remark 2 In fact we expect a stronger conclusion to hold, namely that the set ℋ_{end} ∩ℋ_{diff} has vanishing (n−1)-dimensional Hausdorff measure in ℋ. The rationale for this is as follows: We first note that it is possible to weaken the regularity assumption on 𝒮. For Theorem 1 to hold it is sufficient to require that 𝒮 be a Lipschitz hypersurface that meets the horizon transversely at almost all points of the intersection. This point, taken in conjunction with the fact that the set ℋ_{end} ∩ ℋ_{diff} is necessarily acausal, suggests the possibility of constructing such a Lipschitz hypersurface (or perhaps a countable collection of such Lipschitz hypersurfaces) that contains ℋ_{end} ∩ ℋ_{diff}, at least up to a set of vanishing (n − 1)-dimensional Hausdorff measure. Theorem 1, under this weaker regularity assumption on 𝒮 would then give the strengthened conclusion. Initial efforts to construct such a hypersurface have not been successful.

The set of points where the multiplicity N(p) is large has a more precise structure. To describe this, equip M with an auxiliary complete Riemannian metric σ, and for each p ∈ ℋ let N_{p}^{+} be the set of future pointing σ-unit vectors that are tangent to a generator of ℋ at p. We call such vectors semi-tangents to ℋ. Then the number of vectors in N_{p}^{+} is just the multiplicity N(p) of p. Define

𝒞_{p} := convex cone generated by N_{p}^{+} .

We can measure the size of the set of generators through p by dim(𝒞_{p}) (which is the dimension of the linear span of 𝒞_{p} in T_{p}M). This is a different measure than is N(p); in particular this gives finer information when N(p) = ∞. We also set

ℋ[k] := {p : dim(𝒞_{p}) ≥ k}.  (1.1)

For k = 1, ℋ[k] = ℋ as every point is on at least one generator. For k = 2, ℋ[k] is the set of points of ℋ that are on more than one generator. As dim(𝒞_{p}) is the dimension of the span of N_{p}^{+} and N_{p}^{+} contains N(p) vectors, dim(𝒞_{p}) ≤ N(p). This implies

ℋ[k] ⊆ {p ∈ ℋ : N(p) ≥ k} .

Also, for 1 ≤ k ≤ 3, any k distinct elements of N_{p}^{+} are linearly independent.^{1} Therefore if 1 ≤ k ≤ 3 and N(p) ≥ k then choosing k distinct, and thus linearly independent, elements of N_{p}^{+}, shows that dim(𝒞_{p}) ≥ k. Whence

ℋ[k] = {p ∈ ℋ : N(p) ≥ k} for 1 ≤ k ≤ 3 .

Our next main result, based on the deep results^{2} in [1], is that the sets ℋ[k] are “almost C^{2} submanifolds of dimension n + 1 − k, up to singular sets of lower dimension”.^{3} To make this statement precise, let ℌ^{m} be the m-dimensional Hausdorff measure on M (defined with respect to some Riemannian metric σ on M). Recall, [1, Def. 1.1 p.19] a Borel set Σ ⊂ M is a (ℌ^{m}, m) rectifiable set of class C^{2} iff Σ can be covered, up to a set of vanishing ℌ^{m} measure, by a countable collection of m-dimensional C^{2} submanifolds of M. This definition is independent of the choice of the Riemannian metric σ. Following [1] we will shorten “(ℌ^{m}, m) rectifiable set of class C^{2}” to “C^{2} rectifiable of dimension m”.

1The linear span of two future pointing σ-unit null vectors is a two dimensional timelike subspace and there are only two future pointing null rays in this subspace. So a third future pointing σ-unit vector can not be in the span of the first two.

2The reader is warned that the dimension index k, in ℋ[k], is shifted by one, as compared to that used in Theorem 3 of [1]; compare the remarks following Definitions 1.5 and 1.7 in [1].

3We note a related result of [12], where it is shown that the set {N(p) = 2} is, up to a lower dimensional set, a smooth submanifold of co-dimension two for the horizons considered there.

Theorem 3 For 1 ≤ k ≤ n + 1 the set ℋ[k] is a C^{2} rectifiable set of dimension n + 1 − k. Therefore ℋ[k] has Hausdorff dimension ≤ n + 1 − k.

Using k = 1, and that ℋ[1] = ℋ, this implies that horizons are C^{2} rectifiable of dimension n. As they are also locally Lipschitz graphs they have the further property that ℌ^{n}(ℋ ∩ K) < ∞ for all compact sets K ⊆ M. When k = n + 1 this implies that ℋ[n + 1] is a countable set (cf. Remark 7 below).

2 Proof of Theorem 1

We shall prove the second part of the theorem; the first part follows immediately from the second and the co-area formula. Let, then, 𝒮 be as in Theorem 1; since the result is purely local, without loss of generality we may assume that 𝒮 is the level set {t = 1} of a time function t, with range ℝ, the level sets of which are Cauchy surfaces. We use the constructions and notations of [4], with Σ_{1} = 𝒮 and Σ_{2} = {t = 2}. Let

Ŝ_{1} = 𝒮 ∩ ℋ_{end} ∩ ℋ_{diff} ,

and let A, ϕ be defined as at the beginning of the proof of Theorem 6.1 in [4]. Hence, A is the subset of S_{2} = Σ_{2} ∩ ℋ consisting of those points in S_{2} that are met by the generators of ℋ that meet S_{1} = Σ_{1} ∩ ℋ, and ϕ : A → S_{1} is the map that moves the points of A back along these generators to S_{1}. We can choose the auxiliary Riemannian metric σ on M so that dt has unit length with respect to this metric. Then A is an A_{δ} set as defined by Equation (6.6) of [4], with δ = 1. We set

Â = ϕ^{−1}(Ŝ_{1}) ⊂ A ⊂ S_{2} ;

thus, the points in Â are precisely those points on Σ_{2}∩ℋ the generators through which exit ℋ, when followed to the past, at the differentiable end points on 𝒮.

For i = 1, 2, let ℌ_{h_{i}}^{n−1} denote the (n − 1)-dimensional Hausdorff measure on Σ_{i} with respect to the distance function determined by the induced metric h_{i} on Σ_{i}. By a straightforward extension of the proof of [4, Proposition 6.14] one has for any ℌ_{h_{2}}^{n−1}-measurable subset Ω of A,

∫_{S_{1}} N(p, S_{2})1_{ϕ(Ω)}dℌ_{h_{1}}^{n−1}(p) = ∫_{Ω} J(ϕ)(q) dℌ_{h_{2}}^{n−1}(q) ,  (2.1)

where 1_{U} denotes the characteristic function of the set U, and J(ϕ) is, in a suitably defined sense (cf. [4], Prop. 6.14), the Jacobian of the locally Lipschitz function ϕ.

In Proposition 4 below we show that there exists a ℌ_{h_{2}}^{n−1}-negligible set Â′ ⊂ Â such that J(ϕ)(q) = 0 for all q ∈ Â \ Â′. It then follows that,

Â ⊂ Ω ≡ {q ∈ A : J(ϕ) = 0} ∪ Â′ .  (2.2)

Ω, as defined above, is the union of a ℌ_{h_{2}}^{n−1}-measurable set and a ℌ_{h_{2}}^{n−1}-negligible set, and hence is itself ℌ_{h_{2}}^{n−1}-measurable. Equation (2.1) then shows that ϕ(Ω) is ℌ_{h_{1}}^{n−1}-negligible. Now, since Ŝ_{1} ⊂ ϕ(Ω), the result follows. □

It thus remains to establish the following.

Proposition 4 J(ϕ)(q) = 0 for ℌ_{h_{2}}^{n−1}-almost all q ∈ Â.

Proof: We use the definitions, constructions and notations of the proof of [4, Prop. 6.16]. Thus, let 𝒰 ⊂ Σ_{2} be a coordinate neighborhood of the form 𝒱 × (a, b), with 𝒱 ⊂ ℝ^{n−1} and a, b, ∈ ℝ, in which 𝒰 ∩ N is the graph of a C^{1,1} function g : 𝒱 → ℝ, and in which ℋ ∩ 𝒰 is the graph of a semi–convex function f : 𝒱 → ℝ. Here N = N_{δ} is a locally C^{1,1} hypersurface in Σ_{2} into which A has been embedded. Let prA denote the projection onto 𝒱 of A ∩ 𝒰, thus A ∩ 𝒰 is the graph of g over prA.

Now, let x_{0} ∈ B̂ ∩prÂ, where B̂ is the set of full measure in prA constructed in the proof of [4, Prop. 6.16], and prÂ is the projection onto 𝒱 of Â ∩ 𝒰. Since g is Lipschitz, the graph of g over B̂ ∩ prÂ has full measure in Â. Let q_{0} = (x_{0}, f(x_{0})) ∈ Â be the corresponding point on ℋ ∩ Σ_{2}, thus the generator Γ of ℋ passing through q_{0} exits the horizon at a point p ∈ 𝒮 ∩ ℋ_{diff}. Let Γ̂ be any null geodesic which extends Γ to the past, and let p_{n} be any sequence of points on Γ̂ which are to the causal past of p and which approach p as n tends to infinity. Since Γ̂ is a null geodesic which exits ℋ at p, the p_{n}’s lie to the timelike past I^{−}(ℋ) of ℋ. Thus the integral curve γ_{n} of \frac{∂}{∂t} starting at p_{n} meets ℋ at some point r_{n} ∈ γ_{n} ∩ℋ. One can then construct a causal curve from p_{n} to ℋ ∩ Σ_{2} by following γ_{n} from p_{n} to r_{n}, and any generator of ℋ passing through r_{n}; as (M, g) is globally hyperbolic this generator will necessarily intersect Σ_{2}. It follows that there exists a timelike curve γ̂_{n} ⊂ J^{−}(ℋ) from p_{n} to ℋ ∩ Σ_{2}. By the compactness of the space of causal curves, passing to a subsequence if necessary, the γ̂_{n}’s converge (in a well known sense) to a causal curve γ from p to a point q ∈ ℋ ∩ Σ_{2}. The achronality of ℋ shows that γ is a generator of ℋ passing through p, hence γ = Γ and q = q_{0}.

Now suppose that J(ϕ)(q_{0}) ≠ 0. By the construction of the set B̂, there exists g_{j} ∈ C^{2}(𝒱), approximating g, such that S_{j}, the graph of g_{j}, is a C^{2} hypersurface in Σ_{2} which, in a well defined sense, makes second order contact with ℋ ∩ Σ_{2} at q_{0}. (More precisely, g_{j} and f, as well as their first derivatives, agree at q_{0}, and the second derivative of g_{j} agrees with the so-called second Alexandrov derivative of f at q_{0}.) Since S_{j} is tangent to ℋ ∩ Σ_{2} at q_{0}, the null geodesic Γ̂ is normal to S_{j} at q_{0}. Let ϕ_{j} : S_{j} → 𝒮 be the C^{1} map which moves the points of S_{j} along the family of null geodesics normal to S_{j} which includes Γ̂. Then we have J(ϕ_{j}(q_{0})) = J(ϕ)(q_{0}) ≠ 0, cf. [4, Equation (6.40)]. Define g_{j,ε} ∈ C^{2}(𝒱) by

g_{j,ε}(x) = g_{j}(x) + ε|x − x_{0}|^{2} ,  (2.3)

and let S_{j,ε} be the graph of g_{j,ε}; for ε > 0, S_{j,ε} ⊂ J^{+}(ℋ) and S_{j,ε}\{q_{0}} ⊂ I^{+}(ℋ). Note that as S_{j,ε} is tangent to S_{j} at q_{0}, Γ̂ is normal to S_{j,ε} at q_{0}.

The fact that the Jacobian of ϕ_{j} is nonzero at q_{0} implies that p is not a focal point to S_{j} along Γ̂. Moreover, there can be no focal points to S_{j} along the segment of Γ̂ to the future of p, cf. [4, Lemma 4.15]. It follows that by taking ε small enough and Γ̂ short enough, there will be no focal points to S_{j,ε} along Γ̂. This implies by normal exponentiation that there exists an embedded C^{2} null hypersurface 𝒩_{j,ε} which contains Γ̂ and, by shrinking it if necessary, S_{j,ε}, as well, cf. [4, Prop. A3]. Moreover, there exists a neighborhood 𝒪 of Γ̂ in which 𝒩_{j,ε} is achronal: Indeed, since spacetime is time orientable, 𝒩_{j,ε} is a twosided connected embedded hypersurface in M. As such 𝒩_{j,ε} admits a connected neighborhood O which is separated by 𝒩_{j,ε} (𝒩_{j,ε} ⊂ O, and O \ 𝒩_{j,ε} consists of two components). Then a future directed timelike curve joining points of 𝒩_{j,ε} would be a timelike curve from the future side of 𝒩_{j,ε} to the past side of 𝒩_{j,ε}, which is impossible if the curve remains in O. We conclude that 𝒩_{j,ε} is achronal in O.

Consider now the timelike curves γ̂_{n} ⊂ J^{−}(ℋ) constructed earlier in the proof; since the γ̂_{n}’s converge to Γ there exists n_{0} such that all the γ̂_{n}’s are entirely contained in 𝒪 for n ≥ n_{0}. Moreover, by taking n_{0} larger if necessary, it is clear that each such γ̂_{n} will meet the hypersurface P in M obtained by pushing S_{j,ε} to the past along the integral curves of −\frac{∂}{∂t}. One can then construct a timelike curve from p_{n} to S_{j,ε} contained in 𝒪 by following γ̂_{n} from p_{n} to P, and then an integral curve of \frac{∂}{∂t} to S_{j,ε}. This contradicts the achronality of 𝒩_{j,ε} in 𝒪, and establishes Proposition 4. □

3 Proof of Theorem 3

We start by showing that ℋ has no worse regularity than being the boundary of a convex set:

Proposition 5 For any point p ∈ ℋ there is a coordinate system x^{1}, …, x^{n+1} defined on an open set U ⊆ M so that ℋ ∩ U is given by the graph x^{n+1} = h(x^{1}, …, x^{n}) of a convex function h.

Proof: It is shown in [4, Theorem 2.2] that ℋ is locally the graph of a semiconvex function. That is, there is a coordinate system y^{1}, …, y^{n+1} so that U ∩ ℋ is given by a graph y^{n+1} = u(y^{1}, ⋯, y^{n}) + h(x^{1}, ⋯, x^{n}) where u is C^{∞} and h is convex. Define new coordinates by x^{i} = y^{i} for i = 1, …, n and x^{n+1} = y^{n+1} − u(x^{1}, …, x^{n}). In these coordinates ℋ is given by x^{n+1} = h(x^{1}, …, x^{n}). □

Recall that a convex body in a finite dimensional vector space has a well defined normal cone at each of its boundary points p. One definition of this tangent cone is the set of linear functionals on the vector space such that their restrictions to the body attain a maximum at the point p. The following is an adaptation of this definition to manifolds which allows us to define the tangent cones Nor_{p}(J^{+}(ℋ)) to J^{+}(ℋ) at points p ∈ ℋ = ∂J^{+}(ℋ) in an invariant manner.

Nor_{p}(J^{+}(ℋ)) := {df(p) : f ∈ C^{∞}(M, ℝ) and
f|_{J^{+}(ℋ)} has a local max. at p} .

The following is a special case of a main result^{2} in the paper of Alberti [1, Thm 3 p. 18] adapted to our notation.

Proposition 6 (Alberti) Let B be convex body in an n+1-dimensional vector space. For each k = 1, 2, …, n + 1 let ∂B[k] be the set of points p ∈ ∂B so that dim Nor_{p}(B) ≥ k. Then ∂B[k] is a C^{2} rectifiable set of dimension n + 1 − k. Therefore the Hausdorff dimension of ∂B[k] is less then or equal to n + 1 − k.□

Remark 7 The top and bottom dimensional cases of this are worth remarking on. When k = n + 1 this implies that ∂B[n + 1] is a C^{2} rectifiable set of dimension 0. But then, [1, Theorem 3 and Definition 1.1], ∂B[n + 1] is a countable union of sets of finite ℌ^{0} measure. However the zero dimensional measure ℌ^{0} is just the counting measure, [9, p. 171], so that ℌ^{0}(A) is just the number of points in A. Therefore ∂B[n+1] is countable as it is a countable union of finite sets. Consider, next, k = 1; as dim Nor_{p}(J^{+}(ℋ)) ≥ 1 for all p ∈ ∂B we have ∂B = ∂B[1] and therefore ∂B is a C^{2} rectifiable set of dimension n. Because ∂B is also locally the graph of a Lipschitz function it has the further property that ℌ^{n}(K ∩∂B) < ∞ for all compact sets K. The corresponding fact is not true for ∂B[k] when 2 ≤ k ≤ n − 1 (cf. [1, Thm. 2 p. 18]).

Theorem 3 follows immediately from this, Proposition 5, and

Lemma 8 For each p ∈ ℋ the normal cone Nor_{p}(J^{+}(ℋ)) is given by

Nor_{p}(J^{+}(ℋ)) = {⟨v, ·⟩ : v ∈ 𝒞_{p}} .

Proof: We choose a coordinate system x^{1}, …, x^{n+1} on a open set U containing p as in Proposition 5 so that ℋ ∩ U is given by x^{n+1} = h(x^{1}, …, x^{n}) where h is convex. We may assume that the point p has coordinates (0, …, 0). We also assume that U is of the form V × (a, b) for V an open convex set in = ℝ^{n} and that h takes values in the interval (a, b). Then h is locally Lipschitz and thus the Clarke differential ∂h(0) exists and is a compact convex set of linear functionals on ℝ^{n} ([7, pp. 27–28]). As h is convex ∂h(0) is just the set of sub-differentials to h at 0 in the sense of convex analysis ([7, Prop. 2.2.7 p. 36]). It follows that for q ∈ ℋ ∩ U if we write q = (x, h(x)) with x ∈ V that

Nor_{q}(J^{+}(ℋ)) = {λ(α − dx^{n+1}) : λ ≥ 0, α ∈ ∂h(x)} .

There is another useful description of ∂h(0). Let Ω_{h} be the set of points x in V where the classical derivative dh(x) exists. As h is locally Lipschitz Ω_{h} has full measure in V. Let L_{0} be the set

L_{0} := { lim_{ℓ→∞} dh(x_{ℓ}) : x_{ℓ} ∈ Ω_{h}, x_{ℓ} → 0, and lim_{ℓ→∞} dh(x_{ℓ}) exists.}  (3.1)

of limit points of sequences {dh(x_{ℓ})} of sequences {x_{ℓ}} ⊂ Ω_{h} with x_{ℓ} → 0. Then, [7, Thm 2.5.1 p. 63],

∂h(0) = convex hull of L_{0} .  (3.2)

Letting, as in the introduction, ℋ_{diff} be the set of points where ℋ is differentiable, if x ∈ V and q = (x, h(x)) ∈ ℋ, then x ∈ Ω_{h} if and only if q ∈ ℋ_{diff}. By the theorem of Beem and Królak [2] this is the case if and only if q is on exactly one generator of ℋ. If q = (x, h(x)) ∈ ℋ_{diff} then let v_{q} ∈ N_{q}^{+} be the unique semi-tangent to ℋ at q. Then at q the tangent plane to ℋ can be defined either in terms of dh or in terms of v_{q} to be the set of vectors X ∈ T_{q}(M) so that (dh − dx^{n+1})(q)(X) = 0 or ⟨v_{q}, X⟩ = 0. Thus there is a positive scalar λ so that (dh − dx^{n+1})(q) = λ⟨v_{q}, · ⟩. Therefore the normal cone at q is one dimensional and

Nor_{q}(J^{+}(ℋ)) = {λ⟨v_{q}, · ⟩ : λ ≥ 0} = {λ(dh − dx^{n+1}) : λ ≥ 0} .

It follows from this that if {x_{ℓ}} ⊂ Ω_{h} and q_{ℓ} = (x_{ℓ}, h(x_{ℓ})) then x_{ℓ} → 0 if and only if q_{ℓ} → p and dh(x_{ℓ}) → α if and only if v_{ℓ} → v where ⟨v, · ⟩ = α. Unraveling all this and using (3.1) and (3.2) gives that in order to complete the proof it is enough to show

N_{p}^{+} = { lim_{ℓ→∞} v_{q_{ℓ}} : q_{ℓ} ∈ ℋ_{diff}, q_{ℓ} → p, and lim_{ℓ→∞} v_{q_{ℓ}} exists.} .  (3.3)

Denote the right side of this equation by T_{p}. Then, [4, Lemma 6.4], the set of semi-tangents N^{+} is a closed subset of T (M) and therefore T_{p} ⊆ N_{p}^{+}. If v ∈ N_{p}^{+} then there is a generator c: [0, ∞) → M with c(0) = p, c′(0) = v and parameterized so that it is unit speed with respect to the auxiliary Riemannian metric σ. For each positive integer ℓ, c(1/ℓ) is an interior point of the generator c and thus c(1/ℓ) ∈ ℋ_{diff}. Then v_{c(1/ℓ)} = c′(1/ℓ) and lim_{ℓ→∞} v_{c(1/ℓ)} = lim_{ℓ→∞} c′(1/ℓ) = c′(0) = v. Thus v ∈ T_{p} which yields N_{p}^{+} ⊆ T_{p}. This shows (3.3) holds and completes the proof of the lemma and therefore of Theorem 3. □

4 Application to distance functions and cut loci in Riemannian Manifolds.

Let (S, h) be a connected Riemannian manifold which we do not assume to be complete. Let C ⊂ S be a closed set. Then define the distance function ρ_{C}: S → [0, ∞) by

ρ_{C}(p) := infimum of lengths of smooth cuves in S connecting p to C .

This will be Lipschitz with Lipschitz constant one: |ρ_{C}(p) − ρ_{C}(q)| ≤ d(p, q) where d(p, q) is the Riemannian distance between p and p.

We will see that regularity properties of ρ_{C} and the cut locus of C in S are closely related to the regularity properties of horizons, by looking at the graph of ρ_{C} (cf. Proposition 10). In this setting it is natural to consider the problem even when (S, h) is not complete. For example when S is the interior of a manifold P with boundary then ρ_{C} agrees with the distance from C defined by the infimum of the length of curves from p to C in P so that the results apply to that case as well. Also in the setting of Lorentzian geometry one can use the graphs of functions ρ_{C} to construct examples of horizons regardless of completeness of (S, h).

Let I ⊂ ℝ be an interval (which may be open, closed, or half open). Then a C-minimizing segment on I is a unit speed geodesic γ: I → S so that

ρ_{C}(γ(s)) = s for all s ∈ I .

We emphasize that we do not assume that I contains 0. The Riemannian equivalent of the fact that horizons are null-geodesically-ruled is contained in the following proposition:

Proposition 9 Every p ∈ S \ C is on at least one C-minimizing segment.

Proof: Let U be a convex normal neighborhood of p disjoint from C. For r > 0 sufficiently small the distance sphere S_{r}(p) = {x ∈ M : d(p, x) = r} is contained in U, is compact, and agrees with the geodesic sphere of radius r centered at p. Then, ρ_{C} restricted to S_{r}(p) achieves a minimum at some point q, say. Let γ be the unique minimizing geodesic from q to p. From the choice of q on S_{r}(p), and simple distance function considerations, one has for each x on γ,

d(C, q) + d(q, x) = d(C, x) ,  (4.1)

where d(C, x) = ρ_{C}(x). Since γ is minimizing on each segment, Equation (4.1) implies that γ, when suitably parameterized, is a C-minimizing segment. □

Each C-minimizing segment γ: I → S is contained in a maximal (with respect to the size of the interval of definition) C-minimizing segment and from now on we assume that all C-minimizing segments are defined on their maximal domain. We say that a C-minimizing segment γ: I → S has a cut point iff its interval of definition is of the form [a, b] or (a, b] with b < ∞, in which case p = γ(b) is defined to be the cut point. A C-minimizing segment can fail to have a cut point either because its domain is unbounded, i.e. of the form [a, ∞) or (a, ∞), or because the domain is bounded, say (a, b) but the limit lim_{t↑b} γ(t) does not exist in S. The later condition can not arise when S is complete. When S is complete the domains of C-minimizing segments are all of the form [0, b] or [0, ∞).

The collection of all cut points is the cut locus of C in S and denoted by Cut_{C}. The cut locus Cut_{C} is a subset of S \ C, and the definition here of Cut_{C} agrees with the usual definition when S is complete. For any p ∈ S \ C let N_{C}(p) be the number of C-minimizing segments on which p lies. Then for all p ∈ S \ C we have N_{C}(p) ≥ 1 and it is known that if N_{C}(p) ≥ 2 then p ∈ Cut_{C}. For p ∈ S \ C let

M_{p} := {−γ′(ρ_{C}(p)) : γ is a C-minimizing segment and γ(ρ_{C}(p)) = p}

be the set of the unit vectors at p which are tangent to C-minimizing segments and which point toward C. Then the number of vectors in M_{p} is just N_{C}(p).

Let M = S × (−∞, 0) and give M the Lorentzian metric g = h − dt^{2}. Let ℋ be the graph of −ρ_{C} in M, that is

ℋ := {(x, −ρ_{C}(x)) : x ∈ S \ C} .

We leave the proof of the following to the reader.

Proposition 10 The set ℋ is a future horizon in M. The null generators of ℋ are (up to reparameterization) the curves s ↦ (γ(s), −ρ_{C}(γ(s))), where γ is C-minimizing segment of S and N(p, −ρ_{C}(p)) = N_{C}(p). The set ℋ_{end} coincides with the set {(p, −ρ_{C}(p)) : p ∈ Cut_{C}}. □

The theorem of Beem and Królak [2] that a point p of a horizon is differentiable if and only if N(p) = 1 implies the Riemannian result:

Proposition 11 With notation as above, the point p ∈ S \C is a differentiable point of the distance function ρ_{C} if and only if N_{C}(p) = 1. □

Remark 12 While to best of our knowledge this result has not appeared in the literature on the regularity of Riemannian distance functions, it would be surprising if it were not known, at least in the case of of complete manifolds, to experts. It can also be deduced from general facts about the Clarke differential in non-smooth analysis. Explicitly, it follows from [6, Thm 2.1 (4) p. 251], a result about the generalized gradients of functions that are pointwise minimums of families of smooth functions with appropriate Hessian bounds, that the Clarke differential of ρ_{C} at x ∈ N \ C is

∂ρ_{C}(x) = convex hull of {−⟨u, · ⟩ : u ∈ M_{x}}

(When S is Euclidean space this is [10, Lemma 4.2 pp. 1037–1038]. The extension to complete Riemannian manifolds is not hard.) The function ρ_{C} is semi-convex and a semi-convex function is differentiable at x if and only its Clarke differential at x is a singleton^{4}. Therefore ρ_{C} is differentiable at x if and only if N_{C}(x) = 1. It is also possible to carry out a proof of the Beem and Królak result along these lines.

In [5] horizons in Lorentzian manifolds are constructed that are nondifferentiable on a dense set. Another family of such examples, possessing genericity properties, is given in [3]. When translated into Riemannian terms the examples of [5] imply:

Proposition 13 There exists a closed Lipschitz curve C ⊂ ℝ^{2} such that its distance function ρ_{C} is non-differentiable on a dense subset of the set ρ_{C} ≤ 1 of the unbounded component of ℝ^{2} \ C. □

4The general semi-convex case reduces to the case of convex functions. For a convex function the Clarke differential is the sub-differential in the sense of convex analysis [7, Prop 2.2.7 p. 36] and a convex function is differentiable at point if and only if its sub-differential is a singleton.

In both [3, 5] it is shown how to obtain higher dimensional examples.

Theorem 1 implies

Proposition 14 If C is a closed set in an n-dimensional Riemannian manifold (S, h), then ℌ_{H}^{n}(Cut_{C}) = 0. □

When C is a smooth submanifold of S (for example when C is a point) and the manifold (S, h) is complete then a recent result of Itoh and Tanaka [14, Thm. B p. 22] implies that Cut_{C} has Hausdorff dimension at most n − 1. (When S is two dimensional and C is a point this had been done earlier by Hebda [11] and Itoh [13].) However for arbitrary closed sets the question of the Hausdorff dimension of Cut_{C} is open. In particular it is not known if there is a closed subset C of the Euclidean plane ℝ^{2}, with its usual metric, so that the Hausdorff dimension of Cut_{C} is 2. (Proposition 14 implies ℌ^{2}(Cut_{C}) = 0, but this does not rule out the possibility that the Hausdorff dimension is 2.)

If M = S × (−∞, 0) is given as its auxiliary Riemannian metric σ = h + dt^{2}, then the set of semi-tangents to ℋ at p = (x, −ρ_{C}(x)) is N_{p}^{+} = {(2^{−1/2}u, 2^{−1/2}∂/∂t) : u ∈ M_{x}}. Set

Cut_{C}[k] := {x ∈ S \ C : dim Span{(u, ∂/∂t) : u ∈ M_{x}} ≥ k } .

As in the introduction {x ∈ S \ C : N_{C}(p) ≥ k} ⊆ Cut_{C}[k] and equality holds if 1 ≤ k ≤ 3. Note for k ≥ 2 that Cut_{C}[k] ⊂ Cut_{C}, while for k = 1 we have Cut_{C}[1] = S \ C. The set {x ∈ N \ C : N_{C}(x) ≥ 2} = Cut_{C}[2] is the strict cut locus and for some special choices of C, for example a point or a submanifold, its structure has been studied by several authors (cf. [11, 14] and the references therein). Using that the epi-graph of a distance function has locally positive reach it follows from results of Federer [8, Remark 4.15 (3) p. 447] that each set Cut_{C}[k] is countable rectifiable of dimension n + 1 − k. Theorem 3 allows us to refine this to C^{2} rectifiablity.

Proposition 15 For 1 ≤ k ≤ n + 1 the set Cut_{C}[k] is a C^{2} rectifiable set of dimension n + 1 − k. Therefore Cut_{C}[k] has Hausdorff dimension ≤ n + 1 − k. □

Remark 16 By a theorem of Rademacher the set of non-differentiable points of a Lipschitz function on an n-dimensional manifold has ℌ^{n} measure zero. In the case that the function is a distance function, ρ_{C}, this can be improved. By Proposition 11 the set Cut_{C}[2] is exactly the set of points p ∈ S \ C where the distance function ρ_{C} is not differentiable. Thus, by Proposition 15, the set of non-differentiable points of ρ_{C} is a C^{2} rectifiable set of dimension n − 1 and therefore has Hausdorff dimension at most n − 1.

For each r > 0 let τ_{r}(C) := {p ∈ S : ρ_{C}(p) = r} be the tube of radius r about C. In general these tubes can have singularities and need not be topological hypersurfaces in S. We now look at what the Riemannian versions of [4, Sec. 5] have to say about the regularity of τ_{r}(C) and ρ_{C}. By Proposition 5 the function ρ_{C} is semi-convex on S \ C. Therefore by Alexandrov’s theorem for almost all p ∈ S \ C the function ρ_{C} has second Alexandrov derivatives. This means that in local coordinates x^{1}, …, x^{n} centered at p the function ρ_{C} has a second order Taylor expansion

ρ_{C} = ρ_{C}(p) + dρ_{C}(p)x + \frac{1}{2}D^{2}ρ_{C}(p)(x, x) + o(|x|^{2})

where x = (x^{1}, …, x^{n}), dρ_{C}(p) is a covector at p and D^{2}ρ_{C}(p): T_{p}(S)×T_{p}(S) → ℝ is a symmetric bilinear form. Denote by S_{Al} the set of Alexandrov points of ρ_{C}. At points p ∈ S_{Al} the function ρ_{C} is differentiable and therefore the discussion above yields that N_{C}(p) = 1 and that dρ_{C}(p) = −⟨u, · ⟩ where u ∈ M_{p}. If p ∈ S_{Al} and r = ρ_{C}(p) then the level set τ_{r}(C) has a well defined tangent space T_{p}(τ_{r}(C)) := {X ∈ T_{p}S : dρ_{C}(p)X = 0} and a well defined second fundamental form II at p given by

II_{p}(X, Y) = −D^{2}ρ_{C}(p)(X, Y) for X, Y ∈ T_{p}(τ_{r}(C)) .

Let A_{p}: T_{p}(τ_{r}(C)) → T_{p}(τ_{r}(C)) be the corresponding Weingarten map defined by

⟨A_{p}X, Y ⟩ := II_{p}(X, Y) for X, Y ∈ T_{p}(τ_{r}(C)) .

That is A_{p} is the tensor of type (1, 1) corresponding to the tensor II of type (0, 2) and A_{p} is a self-adjoint linear map. With the choice of signs here, when S = ℝ^{n} and C is the origin (so that τ_{r}(C) is the sphere of radius r) for p ∈ τ_{r}(C) we have A_{p} = −\frac{1}{r}I. Let the Riemannian connection on S be denoted by ∇ and let R be the curvature tensor of S with signs chosen so that R(X, Y)Z = (∇_{X}∇_{Y} −∇_{Y} ∇_{X} −∇_{[X,Y]})Z. For U ∈ T_{p}M define a linear map R_{U}: T_{p}S → T_{p}S be R_{U}X = R(X, U)U. The following summarizes the Riemannian versions of Proposition 3.1, Proposition 3.5, Theorem 5.1 and Theorem 5.6 of [4].

Theorem 17 Let γ: I → S by a C-miminizing segment with γ(0) ∈ C. If γ(t_{0}) ∈ S_{Al} for some t_{0} ∈ I, then γ(t) ∈ S_{Al} for all t ∈ I_{0} := {t_{0}} ∪ (I ∩ [t_{0}, ∞))^{∘} (where J^{∘} is the interior of J ⊂ ℝ). The Weingarten maps A_{γ(t)} of the tubes τ_{ρ_{C}(γ(t))}(C) vary smoothly on the interval I_{0} and satisfy the usual Riccati equation for parallel hypersurfaces:

∇_{\frac{d}{dt}} A_{γ(t)} = A_{γ(t)}^{2} + R_{γ′(t)} .

Moreover for all r ∈ (0, ∞) the tube τ_{r}(C) has locally finite ℌ^{n−1}-dimensional measure and for almost all r ∈ (0, ∞) there is a set P ⊂ τ_{r}(C) with ℌ^{n−1}(P) = 0 so that every C-miminizing segment γ: I → S which meets τ_{r}(C) \ P will have γ(t) ∈ S_{Al} for t ∈ I^{∘}. □

In the terminology of [4] “Alexandrov points propagate to the past along generators” of horizons. In the Riemannian setting Alexandrov points of a distance function ρ_{C} propagate away from C along C-miminizing segments. The last sentence of Theorem 17 implies loosely that almost every C-miminizing segment is an Alexandrov segment in the sense that all of its points other than its endpoints are in S_{Al}.

References

[1] G. Alberti. On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations, 2(1):17–27, 1994.

[2] J. K. Beem and A. Królak. Cauchy horizon end points and differentiability. J. Math. Phys., 39(11):6001–6010, 1998.

[3] R. J. Budzyński, W. Kondracki, and A. Królak. On the differentiability of Cauchy horizons. J. Math. Phys., 40(10):5138–5142, 1999.

[4] P. T. Chruściel, E. Delay, G. J. Galloway, and R. Howard. Regularity of horizons and the area theorem. To appear in Annales H. Poincaré, gr-qc/0001003, 2000.

[5] P. T. Chruściel and G. J. Galloway. Horizons non-differentiable on a dense set. Comm. Math. Phys., 193(2):449–470, 1998.

[6] F. H. Clarke. Generalized gradients and applications. Trans. Amer. Math. Soc., 205:247–262, 1975.

[7] F. H. Clarke. Optimization and nonsmooth analysis. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1990.

[8] H. Federer. Curvature measures. Trans. Amer. Math. Soc., 93:418–491, 1959.

[9] H. Federer. Geometric measure theory. Springer-Verlag New York Inc., New York, 1969. Die Grundlehren der mathematischen Wissenschaften, Vol. 153.

[10] J. H. G. Fu. Tubular neighborhoods in Euclidean spaces. Duke Math. J., 52(4):1025–1046, 1985.

[11] J. J. Hebda. Metric structure of cut loci in surfaces and Ambrose’s problem. J. Differential Geom., 40(3):621–642, 1994.

[12] S. Husa and J. Winicour. Asymmetric merger of black holes. Phys. Rev. D (3), 60(8):084019, 13, 1999. gr-qc/9905039.

[13] J. Itoh. The length of a cut locus on a surface and Ambrose’s problem. J. Differential Geom., 43(3):642–651, 1996.

[14] J. Itoh and M. Tanaka. The Lipschitz continuity of the distance function to the cut locus. Trans. Amer. Math. Soc., 353(1):21–40, 2000.