ON THE CURVATURE OF NONREGULAR SADDLE SURFACES IN THE HYPERBOLIC AND SPHERICAL THREE-SPACE

ON THE CURVATURE OF NONREGULAR SADDLE SURFACES IN THE HYPERBOLIC AND

SPHERICAL THREE-SPACE

DIMITRIOS E. KALIKAKIS Received 9 November 2001

This paper proves that any nonregular nonparametric saddle surface in a three- dimensional space of nonzero constant curvatureκ, which is bounded by a recti- fiable curve, is a space of curvature not greater thanκin the sense of Aleksandrov.

This generalizes a classical theorem by Shefel’ on saddle surfaces inE³. 1. Introduction

The class of saddle surfaces is dual to the class of convex surfaces. A surface in a Euclideann-space is said to be a saddle surface if it is impossible to cut offa crust by any hyperplane. In contrast to the theory of convex surfaces, the results in the theory of saddle surfaces are in many respects far from complete.

One of the central problems in this area is the study of the intrinsic geometry.

Although it is known that the Gaussian curvature of a regular saddle surface in Eⁿis nonpositive, it remains an open question whether the intrinsic curvature of any nonregular saddle surface inEⁿis nonpositive. An affirmative answer has been given by Shefel’ [4,7] whenn=2 (for any simply connected saddle surface), and whenn₌3 (for any nonparametric saddle surface). The answer is still not known forn >3.

In order to describe our results on saddle surfaces, first we need to intro- duce some terminology. Then-dimensionalκ-spaceSⁿ_κ(κ-plane forn₌2) is the hyperbolic spaceHⁿ_κ forκ <0, the Euclidean spaceEⁿforκ=0, and the upper open hemisphereSⁿ+(κ^−1/2) ofEⁿ⁺¹of radiusκ^−1/2with the induced metric, when κ >0. EverySⁿ_κis a Riemannian simply connected manifold of constant sectional curvatureκsuch that any pair of points can be joined by a unique geodesic seg- ment. Notice thatSⁿ_κis a complete space only ifκ≤0.

A nonparametric surface in the Beltrami-Klein model ofH³_κis a continuous functionz₌ f(x, y), providedx²+y²+z²<₋1/κ. A nonparametric surface in

Copyright©2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:3 (2002) 113–123

2000 Mathematics Subject Classification: 53C45, 53A35, 52B70 URL:http://dx.doi.org/10.1155/S1085337502000799

S³+(κ^−1/2) is a surface represented in the formr(x,y)₌(x/a, y/a, f(x,y)/a, κ^−1/2/a), wherea=(1 +κx²+κy²+κ f²(x, y))^1/2 and f is a continuous function of two variables. The principal result of this work is the following.

Theorem1.1. If a nonparametric saddle surface inS³_κ (κ₌0) is bounded by a rectifiable curve, then it is a space of curvature bounded from above byκin the sense of Aleksandrov.

The converse ofTheorem 1.1does not hold asExample 6.1shows. The proof of Theorem 1.1is based on the possibility of approximating a nonparametric saddle surface inS³_κby saddle polyhedra (Lemma 5.2) and on a characterization of spaces of curvature bounded from above in the sense of Aleksandrov due to Reshetnyak (Lemma 5.1). In higher dimensions the possibility of such an ap- proximation is still not known even in the Euclidean case (see [4, page 59]).

Saddle surfaces inS³_κ (κ₌0) can be defined in a similar way as inE³, that is, by means of the operation of cutting offcrusts byκ-planes. Instead of this definition, we introduce an equivalent coordinate-free definition using only the geodesic structure ofS³_κ.

InSection 2, we review the definition of a metric space of curvature bounded from above in the sense of Aleksandrov. InSection 3, we present the general- ized definition of a saddle surface in an arbitrary geodesically connected space (Definition 3.1,Theorem 3.4). InSection 4, we determine the curvature condi- tion that a saddle polyhedron inS³_κsatisfiesProposition 4.4and inSection 5, we give the proof ofTheorem 1.1.

2. Metric spaces of curvature bounded from above in the sense of Aleksandrov

A notion of curvature of metric spaces can be defined by comparing triangles in a metric space with the corresponding model triangles in theκ-plane with sides of the same length. The definition is due to Aleksandrov [1] and the curva- ture is usually referred to as the curvature in the sense of Aleksandrov. Aleksan- drov’s spaces are a natural generalization of Riemannian manifolds but they are of much more general nature. For more details, see [2,3].

AnRκdomain, abbreviated byRκ, is a metric space satisfying the following axioms.

Axiom 1. Any two points inRκcan be joined by a geodesic segment.

Axiom 2. Ifκ >0, then the perimeter of each triangle inRκis less than 2π/^√κ.

Axiom 3. Each triangle in Rκ has nonpositive κ-excess, that is, for the angles α, β, γof a triangleABC

α+β+γ₋α_κ+β_κ+γ_κ≤0, (2.1)

whereα_κ, β_κ, γ_κare the corresponding angles of a triangleA^κB^κC^κon theκ-plane with sides of the same length asABC.

Another term for anRκ domain is a CAT(κ) space. It is evident that anyκ- space is anRκdomain. A space of curvature bounded byκfrom above in the sense of Aleksandrov is a metric space, each point of which is contained in some neighborhood of the original space, which is anRκdomain.

3. Saddle surfaces

Nonregular saddle surfaces inEⁿ. A (parametrized) surface f inEⁿis any con- tinuous mappingf :D_→Eⁿ, whereDdenotes the closed unit disk on the plane.

We say that a hyperplaneP with equationa1x1+···+anxn=bcuts offa crust from the surface f if among the connected components (maximal connected subsets) of f⁻¹(f(D)P) there is one with positive distance from the boundary ofD. It is clear that if U is such a component, thenU is an open set and the set f(U), which is called a crust, is contained in one of the two open half-spaces that the hyperplanePdefines. We always assume thatf(U)_⊂P⁺and f(∂U)_⊂P, whereP⁺is the half-space determined bya1x1+···+anxn> b.

A surface f inEⁿis said to be a saddle surface if it is impossible to cut off a crust from it by any hyperplane (see [4]). Notice that saddle surfaces are, by definition, compact surfaces. The class ofC²saddle surfaces inE³coincides with the class of surfaces of nonpositive Gaussian curvature.

Nonregular saddle surfaces in metric spaces. Let (M, d) be a geodesically con- nected metric space, andDthe closed unit disk on the plane. A (parametrized) surface f in a metric spaceMis any continuous mapping f :D→M. The con- vex hull of a subsetA, denoted by conv(A), is defined as the union of all sets G⁽ⁿ⁾(A), withG⁽⁰⁾(A)₌A,G⁽¹⁾(A) is the union of all geodesic segments between points ofA, andG⁽ⁿ⁾(A)=G⁽¹⁾(G⁽ⁿ⁻¹⁾(A)) for anyn >1.

Definition 3.1. A surface f in a geodesically connected spaceM is said to be a saddle surface if

f(intγ)⊂convf(γ) (3.1)

for every Jordan curveγ_⊂Dhaving positive distance from the unit circle.

Theorem 3.4below shows the equivalence ofDefinition 3.1with the classical one in the case of a Euclidean space. In order to prove it we need the following two elementary lemmas.

Lemma3.2. LetD1, . . . , Dmbe closed disks in the plane such that^m_i=1Diis a con- nected set. Then given anε >0, there exists a Jordan plane curveγwith the follow- ing properties:

(a)γconsists of a finite number of circular arcs each of which is a part of the boundary of someD_i(i=1, . . . , m), whereD_iis a closed disk with the same center asD_iand its radius is(1 +λ_i)times the radius ofD_i, where0≤λ_i< ε;

(b)^m_i=1D_i⊂^m_i=1D_i⊂intγ.

We sketch the proof ofLemma 3.2. The claim is obvious form₌1. Suppose that the claim is true for somem≥1. Let ε >0 andD1, . . . , D_m, D_m+1 be m+ 1 closed disks in the plane with^m+1_i=1 Di a connected set. We group the disks D1, . . . , D_mintokgroups so that the union of each such group is a connected set.

Then we apply the inductive assumption for each one of these groups and we get kJordan curvesγ1, . . . , γkandmnew closed disksD₁, . . . , D_m. IfDm+1touches any one of the disksD₁, . . . , D_m, then we slightly enlargeD_m+1to a new oneD_m+1that does not touch any one of them. Then the desired Jordan curve is the boundary of the unbounded component ofE²\(intγ1∪···∪intγk∪D_m+1).

Letδ >0. The closure of a bounded connected set in the plane can be covered by a finite number of open disks of radiusδ/4 the union of which is a con- nected set. Therefore, the Jordan plane curve thatLemma 3.2ensures for the corresponding closed disks and for the positive numberε₌δ/4 satisfies the two conditions of the following lemma.

Lemma3.3. LetUbe a bounded, connected set in the plane and letδbe a positive number. Then there exists a Jordan plane curveγsuch that(i)γ⊂

y∈∂UD(y, δ), and(ii) ¯U_⊂intγ, whereD(y, δ)denotes the open disk of radiusδcentered aty.

The following theorem justifies our definition of a saddle surface.

Theorem3.4. If f is a surface inEⁿthen the following are equivalent:

(a)it is impossible to cut offa crust from f by any hyperplane,

(b) f(intγ)⊂conv(f(γ))for every Jordan curve γ⊂Dwhich has a positive distance from the unit circle.

Proof. (a)⇒(b). Suppose, contrary to the claim, that there exist a Jordan curve γ⊂Dhaving a positive distance from the unit circle, and a pointa∈int(γ) so that f(a)∈ conv(f(γ)). We can separate the convex set conv(f(γ)) from the point f(a) by a hyperplaneP with f(a)_∈P⁺and conv(f(γ))_⊂P⁻, whereP⁺ andP⁻are the two open half-spaces the hyperplaneP defines. IfV is the con- nected component of f⁻¹(P⁺) that contains the pointa∈ int(γ), thenV does not intersect the curveγsince f(γ)_⊂P⁻. SoV _⊂int(γ) and therefore the dis- tance ofVfrom the unit circle is positive. Thus, the hyperplanePcuts offa crust from f, a contradiction.

(b)⇒(a). Suppose that a hyperplanePcuts offa crust from f. Then f(U)_⊂ P⁺ and f(∂U) _⊂P for some open connected subset U of D having positive distance from the unit circle. Letε=max{dist(x, P) :x ∈ f( ¯U)}>0. Since f is a uniformly continuous function, there is aδ1 > 0 such that f(D(y, δ)) _⊂ B(f(y), ε/2) for allδ_∈(0, δ1) withD(y, δ)_⊂D, whereB(f(y), ε/2) denotes the

n-dimensional ball of radiusε/2 centered at f(y). Since the distance ofUfrom the unit circle is positive, chooseδ∈(0, δ1) so that_y∈∂UD(y, δ)⊂D. Then

f(γ)_⊂f





y∈∂U

D(y, δ)



⊂

y∈∂U

fD(y, δ)

⊂

y∈∂U

B f(y),ε 2

⊂

z∈P

B z,ε 2

.

(3.2)

Therefore, f(γ)⊂ {p+tn:p∈Pand −ε/2≤t≤ε/2}, wherenis a unit normal vector to the hyperplaneP. So f(intγ)⊂conv(f(γ))⊂ {p+tn:p∈Pand−ε/2≤ t_≤ε/2_}therefore, byLemma 3.3(ii), f( ¯U)_{⊂ {}p+tn:p_∈Pand−ε/2_≤t_≤ε/2_}

which contradicts the choice ofε.

Definition 3.5. LetM1, M2be two metric spaces. The mappingϕ:M1→M2is called a geodesic mapping if the image of any geodesic segment inM1underϕis a geodesic segment inM2.

Example 3.6. For anyκ_∈Rthere exists a mappingϕ:S³_κ→E³such that bothϕ andϕ⁻¹are geodesic mappings.

Proof. The assertion is trivial whenκ₌0. Whenκ <0 consider the Beltrami- Klein model ofH³_κ. Since geodesic segments in the Beltrami-Klein model ofH³_κ coincide with the Euclidean line segments, the inclusion mappingϕ:H³_κ→E³ withϕ(x)₌x and its inverse are geodesic mappings. In the case when κ >0 consider the central projectionϕ:S³+(κ^−1/2)→E³defined by

ϕx1, x2, x3, x4

=κ^−1/2 x1

x4,x2

x4,x3

x4

. (3.3)

The central projection takes a pointx onS³₊(κ^−1/2) to the intersection of the hyperplane{x4=κ^−1/2_{} ≡}E³with the straight line through the pointxand the origin ofE⁴. Under the mappingϕgreat circles go to straight lines and vice versa.

Therefore, bothϕandϕ⁻¹are geodesic mappings.

Proposition3.7. Let M1, M2be two metric spaces,ϕ:M1 →M2 be a geodesic mapping, and f :D_→M1be a saddle surface inM1. Thenϕ_◦f is a saddle surface inM2.

Proof. It follows directly by the definition of saddle surfaces and convex hull.

4. Curvature of saddle polyhedra inS³_κ

In order to determine the curvature condition that saddle polyhedra inS³_κ sat- isfy, we need to estimate the total angle at any point of such a polyhedron. All arguments in this section can be trivially generalized to higher dimensions.

A surface inS³_κ, defined over a domain in the Euclidean plane bounded by a simple closed polygonal line, is called a polyhedron if it can be partitioned into

a finite number ofκ-plane triangles intersected only at the boundaries. In order to estimate the total angle at a point of a saddle polyhedron inS³_κ, we need the following two lemmas.

Lemma4.1. IfA1, A2, A3, A4, andOare points inS³_κsuch thatObelongs to the convex hull ofA1, A2, A3,andA4, then

A1OA2+A2OA3+A3OA4+A4OA1≥2π. (4.1) Proof. First letκ₌0.O_∈conv{A1, A2, A3, A4}implies that there exists a pointD on the line segmentA3A4and a pointBon the line segmentA1Dsuch thatOlies on the line segmentA2B. Because of triangle inequality and sinceA1, A2,andD are coplanar, we have

A1OA2+A2OA3+A3OA4+A4OA1

=A1OA2+ (A2OA3+A3OD) + (DOA4+A4OA1)

≥A1OA2+A2OD+DOA1

=2π.

(4.2)

Since in the Beltrami-Klein model ofH³_κgeodesic segments are Euclidean line segments, the proof in the hyperbolic case is exactly the same as in the Euclidean case. In the hemisphereS³+(κ^−1/2) we follow the same steps as in the Euclidean case. Equality (4.2) holds because the images ofA1, A2,andDunder exp⁻¹_O are

coplanar.

Lemma4.2. LetO, B, andA1, A2, . . . , A_kbe points inS³_κ. IfBbelongs to the convex hull ofA1, A2, . . . , A_k, then

AOB +BOC _≤AOA1+A1OA2+···+A_k−1OA_k+A_kOC (4.3) for anyA, CinS³_κ.

Proof. Apply induction onkand the angle triangle inequality.

Proposition 4.3. The total angle at any point of a saddle polyhedron in S³_κ is greater than or equal to2π.

Proof. LetObe a point on a saddle polyhedron inS³_κ. Then, byDefinition 3.1, there are points A1, A2, . . . , Ak on the polyhedron such that O ∈ conv{A1, A2, . . . , A_k}. We will prove that

A1OA2+A2OA3+···+A_k−1OA_k+A_kOA1≥2π. (4.4) Ifk₌3 then relation (4.4) obviously holds as an equality. Letk >3, then there exists a point B _∈ conv_{A3, . . . , A_k−1} such that O _∈ conv_{A1, A2, B, A_k}. By

Lemma 4.2

A2OB+BOAk≤A2OA3+A3OA4+···+A_k−2OA_k−1+A_k−1OAk (4.5) and, byLemma 4.1,

A1OA2+A2OB+BOA_k+A_kOA1≥2π. (4.6) Therefore,

A1OA2+A2OA3+···+A_k−1OA_k+A_kOA1≥2π. (4.7) Proposition4.4. Any saddle polyhedron in a space of constant curvatureκis a space of curvature bounded from above byκin the sense of Aleksandrov.

Proof. A necessary and sufficient condition for a locally geodesically connected spaceM with intrinsic metric to be a space of curvature _≤κ in the sense of Aleksandrov is

κint(p)≤κ ∀p∈M (4.8)

withκint(p) to be the intrinsic curvature ofMatp, defined by κint(p)₌lim

᐀→p

δ(᐀)

S(᐀). (4.9)

The limit is taken over all nondegenerate geodesic triangles᐀inMthe vertices of which approach the pointp.δ(᐀) is the excess of᐀, that is,δ(᐀)=α+β+γ−π withα, β, γthe angles of᐀, andS(᐀) denotes the area of᐀. This characteriza- tion of spaces of curvature bounded from above is due to Aleksandrov [1].

LetPbe a polyhedron in a space of constant curvatureκandpbe a point on P. If the pointpis not a vertex, then by the Gauss-Bonnett formula,

δ(᐀)=

᐀κ dS=κS(᐀) (4.10)

soκint(p)=κ. Letpbe a vertex ofPwhich belongs to the interior of the triangle

᐀. Suppose that the edges ofP, starting at the vertexp, intersect the sides of᐀ intoNpoints. Joiningpwith theseNpoints and the three vertices of the triangle

᐀, we can constructN+ 3 triangles each of which lies on only one face ofPwith the singleton{p}to be their intersection. Applying the Gauss-Bonnett formula to each of them, we haveδ(᐀1) +δ(᐀2) +···+δ(᐀N+3)=κS(᐀) and therefore, if α, β, γare the three angles of᐀, then

α+β+γ₋(N+ 3)π+ [total angle atp] +Nπ₌κS(᐀). (4.11) Hence,δ(᐀)₌κS(᐀) + [2π₋total angle atp]. But, byProposition 4.3, the total

angle at any vertex of a saddle polyhedron is greater than or equal to 2π. There-

foreδ(᐀)≤κS(᐀), and henceκint(p)≤κ.

5. Curvature of saddle surfaces inH³_κandS³+(κ^−1/2)

In this section we proveTheorem 1.1. To do so we need the concept of aPκ

domain and the following two lemmas.

A geodesically connected space with intrinsic metric is said to be aPκdomain if for any triangle contained inPκ, whose perimeter is less than 2π/^√κ,κ >0, theκ-excess is nonpositive. It is clear thatPκdomains andRκdomains coincide inS³_κ.

Lemma5.1 (see [6]). A geodesically connected spaceM with intrinsic metric is a Pκdomain if and only if for any closed rectifiable curveᏸinMthere exists a convex domainV inS²_κwith bounding curveᏺand a mappingϕ:V→Msuch that(i)ϕ is a nonexpanding mapping, that is,dM(ϕ(x), ϕ(y))≤d_S²_κ(x, y)for allx, y∈Vand (ii)ϕmapsᏺontoᏸtranslating each arc ofᏺonto an arc ofᏸof the same length.

Lemma5.2. Any nonparametric saddle surface inS³_κ(κ₌0) can be approximated uniformly by a sequence of saddle polyhedra with the lengths of their bounding curves convergent to the length of the bounding curve of the saddle surface.

Proof. The caseκ=0 is due to Shefel’ [4,7]. Letκ=0 andϕthe geodesic map- ping fromS³_κintoE³ insured byExample 3.6. It is not difficult to see that the restriction ofϕ:S³_κ→E³to a compact set is a bi-Lipschitz mapping.

Comment 1. Let (g_{i j}(κ)) be the 3×3 positive definite symmetric matrix that the coefficients of the first fundamental form ofS³_κdefine. Eachgi j(κ) is a polyno- mial inx1, x2, x3depending onκ. Assume thatλ1andλ2are the minimum and maximum eigenvalue of (g_{i j}(κ)), respectively. Then, sinceϕis restricted on a compact set, there are positive constantsk1, k2such that 0< k1≤λ1≤λ2≤k2

and

k1≤ 3

i j=1gi jdxidxj

dx₁²+dx²₂+dx₃^2≤k2, (5.1) that is,

k1

dx²₁+dx₂²+dx²_3≤ds²_S3 κ≤k2

dx²₁+dx²₂+dx²₃. (5.2) Equation (5.2) completes the proof of our assertion forκ <0. Letκ >0. On a compact subset ofx₁²+x²₂+x²₃<1/κthe element of lengthds²of the coordinate system (3.3), wherex4=(1/κ₋x²₁₋x²₂₋x₃²)^1/2, satisfies the inequality

c1

dx₁²+dx²₂+dx²₃≤ds²≤c2

dx²₁+dx₂²+dx²₃ (5.3) for some positive constantsc1, c2. Therefore, for anyκ ₌0, the restriction of ϕ:S³_κ→E³to a compact set is a bi-Lipschitz mapping.

We are now ready to complete the proof of the lemma.

Let f be a nonparametric saddle surface inS³_κ and let _{P_n:n_∈N}be the sequence of Euclidean saddle polyhedra approximating the nonparamertic sad- dle surfaceϕ(f). Then,ϕ⁻¹(P_n) is the desired sequence.

Remark 5.3. The fact that the geodesic mappingϕ:S³_κ→E³is bi-Lipschitz on the compact sets has two important consequences; there are positive constants k1, k2depending on the compact set such that for any curveγand surface f in the compact setk2(γ)≤(ϕ◦γ)≤k1(γ) andk²₂S(f)≤S(ϕ◦f)≤k²₁S(f), where denotes length andSdenotes the Lebesgue area (see [5]).

It is a well-known property of a two-dimensional one connected Euclidean surface with nonpositive curvature that its intrinsic diameter does not exceed the half of the length of its bounding curve. Hence, byProposition 4.4,Remark 5.3, andLemma 5.2it follows that any pair of points on the graph᏿of a nonpara- metric surface inS³_κ can be joint by a rectifiable curve on᏿. Therefore, if we consider᏿as a metric space with distance between two points the minimum length of the curves lying on᏿and joining those points, then᏿is a space with an intrinsic metric.

Proof ofTheorem 1.1. Let᏿be the graph of a nonparametric saddle surface inS³_κ bounded by a rectifiable curve. To show that᏿is a space of curvature not greater thanκin the sense of Aleksandrov it suffices to prove that for any curveᏸon᏿ of lengththere exists a nonexpanding mappingϕas described inLemma 5.1.

LetWbe the neighborhood on᏿with boundary curve a given curveᏸof length . Consider W as a space with intrinsic metric induced by the metric of S³_κ. Construct a sequence of saddle polyhedra P_n convergent to W uniformly, so that ifn is the length of the boundary curveᏸnofPnthen limn→∞n=. By Proposition 4.4eachP_n, as a space with intrinsic metric, is a space of curva- ture bounded from above byκ in the sense of Aleksandrov. For the boundary curveᏸn of any saddle polyhedronPn construct, usingLemma 5.1, a nonex- panding mappingϕ_n:V_n→P_nsuch that (a)d_n(ϕ_n(x), ϕ_n(y))_≤d_S²_κ(x, y) for all x, y_∈V_n, and (b)ϕ_nmapsᏺnontoᏸntranslating each arc ofᏺnonto an arc ofᏸn of the same length, whereVnis a convex domain inS²_κ with bounding curveᏺn, anddnis the intrinsic metric ofPn. Since the lengths ofᏺnare uni- formly bounded we can assume, without loss of generality, that the sequence of convex domainsVnconverges to a convex domainV with bounding curveᏺin the Hausdorffsense. The mappingϕ:V→Wdefined byϕ(x)=limn→∞ϕn(xn), where_{x_{n ∈}V_n:n₌1,2, . . ._}is a sequence convergent tox, is a well-defined mapping because

d_nϕ_nx_n, ϕ_ny_n≤d_S²_κx_n, y_{n ∀}n₌1,2, . . . . (5.4) Taking lim inf on both sides of the above inequality and using the semi-continu- ity of length, we have thatϕis a nonexpanding mapping. Condition (b) and the

choice ofVandᏸnimply thatϕmapsᏺontoᏸtranslating each arc ofᏺonto an arc ofᏸof the same length. This completes the proof ofTheorem 1.1.

6. Remarks

(1) The curvature condition in Theorem 1.1 is a necessary but not sufficient condition for a nonparametric surface with rectifiable bounding curve to be sad- dle, as the following elementary example indicates. Similar examples inH³_κand S³+(κ^−1/2) can be obtained by considering the geodesic mappings ofExample 3.6.

Example 6.1. Consider the polyhedron P defined by the points A1(0,0,0), A2(1,0, ε),A3(0,0,1),A4(0,1, ε),A5(−1,0, ε), andA6(0,−1, ε), whereεis any suf- ficiently small positive number. The bounding curve ofP is the polygonal line A2A3A4A5A6A2 and the only vertex is the pointA1. Ifθ(ε) is the total angle of Pat the vertexA1, then limε→0θ(ε)=5π/2>2π. The intrinsic curvature ofPis, by definition, zero everywhere except the vertexA1where it is equal to 2π₋θ(ε).

Therefore, for sufficiently smallε >0 the intrinsic curvature ofPis nonpositive.

But on the other hand, for any suchεthe polyhedronPis not a saddle since we can cut offa crust about the vertexA1.

(2) In [8] it is proved that any simply connected saddle surface inE³satisfies the isoperimetric inequalityαS₋²_≤0 for some positive constantα. Therefore, byRemark 5.3, any simply connected saddle surface inS³_κsatisfies the isoperi- metric inequality βS₋² _≤0 for some positive constant βdepending on the distance of the surface from the boundary of the space. Hence, any simply con- nected saddle surface inS³_κ with rectifiable bounding curve has finite area. On the other hand, in [5] it is proved that at each point of a surface in E³ with finite Lebesgue area there are arbitrarily small neighborhoods bounded by recti- fiable curves. ByRemark 5.3, this is also true in any space of constant curvature.

Therefore,Theorem 1.1can be strengthened as follows.

Theorem6.2. If a saddle surface inS³_κ(κ=0) has a rectifiable bounding curve, and in a neighborhood of each of its points it is nonparameric, then it is a space of curvature bounded from above byκin the sense of Aleksandrov.

(3) Since any simply connected saddle surface inE²can be approximated by a sequence of saddle polyhedra (see [8]), one can easily derive the following theo- rem by applying arguments similar to what we used in the proof ofTheorem 1.1.

Theorem6.3. Any simply connected saddle surface inS²_κ(κ₌0)has a curvature not greater thanκin the sense of A. D. Aleksandrov.

Acknowledgments

The author expresses his gratitude to Professor Igor G. Nikolaev for his useful discussions on this paper. This work was partially supported by the Alexander S.

Onassis Public Benefit Foundation.

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[7] S. Z. Shefel’,On the intrinsic geometry of saddle surfaces, Sibirsk. Mat. ˇZ.5(1964), 1382–1396.

[8] ,On saddle surfaces bounded by a rectifiable curve, Dokl. Akad. Nauk SSSR 162(1965), 294–296, translated from Dokl. Akad. Nauk SSSR162(1965), 294–

296.

Dimitrios E. Kalikakis: Department of Mathematics, University of Illinois at Urbana-Champaign,1409West Green Street, Urbana, IL61801, USA

Current address: Department of Applied Mathematics, University of Crete, Heraklion,714-09, Crete, Greece

E-mail address:[email protected]