Generic Uniqueness of Least Area Planes in Hyperbolic Space

a rX iv:mat h /4866v1[mat h.GT]4A ug24GENERIC UNIQUENESS OF LEAST AREA PLANES IN HYPERBOLIC SPACE BARIS COSKUNUZER A BSTRACT .We study the number of solutions of asymptotic Plateau problem in H 3.By using the analytical results in [Co1]and some topological arguments,we show that there exist an open dense subset of C 3,µJordan curves in S 2∞(H 3)such that any curve in this set bounds a unique least area plane in H 3.1.I NTRODUCTION The asymptotic Plateau problem in H 3is introduced to the mathematical litera-ture by Michael Anderson in his seminal paper [A1].He proved that for any Jordan curve in asymptotic boundary of hyperbolic space S 2∞(H 3),there exists an embed-ded least area plane in H 3spanning that curve by using geometric measure the-ory methods.Then,Hardt and Lin studied the asymptotic regularity of least area surfaces in hyperbolic space in [HL],[Li1].Also in [Ga],Gabai solved asymp-totic Plateau problem by using topological arguments developed by Hass and Scott [HS].Lang generalized Anderson’s methods to solve the problem in Hadamard Gromov hyperbolic spaces [L].The author solved the problem for Gromov hyper-bolic spaces with cocompact metric by generalizing Gabai’s techniques in [Co1].On the number of least area planes spanning a given curve,there are a few results so far.Anderson showed that if the curve in S 2∞(H 3)bounds a convex domain,then there exist a unique least area plane spanning that curve in [A1].Then,Lin generalized this result to the curves bounding star shaped domains in S 2∞(H 3)in [Li1].Anderson also gave an example of a curve bounding infinitely many least area surfaces in H 3,[A2].Recently,the author showed a generic finiteness result for C 3,µcurves in [Co1].In this paper,we improve the generic finiteness result of the previous paper [Co1]to a generic uniqueness result for more general class of curves.In [Co1],we studied the asymptotic Plateau problem by using global analysis methods.We first looked at the space of minimal maps from D 2to H 3with C 3,µasymptotic data.We iden-tified these maps with conformal harmonic maps,and considered this space as a subspace of the space of harmonic maps.Then,we identified these harmonic maps with their boundary parametrizations,by using Li and Tam’s results [LT1],[LT2].Then by using Tomi and Tromba’s techniques in [TT],we proved that the boundary restriction map from minimal maps to their boundary parametrizations is Fredholm12BARIS COSKUNUZERof index0.By using the boundary regularity results of Hardt and Lin in[HL]to get compactness,we showed that for a generic set of curves(except a set offirst category),there arefinitely many least area planes spanning it.The short outline of this paper is the following:We start with the results in the previous paper,and consider the boundary restriction map from minimal maps to their boundary parametrizations whose derivative is isomorphism for the generic curves.By using inverse function theorem,wefind a neighborhood of a least area plane in the preimage mapping homeomorphically to a neighborhood of the generic curve.By using these neighborhoods,we show existence of foliated neighborhood for the least area plane,which implies uniqueness of the least area plane.Then we show same result for any curve in a neighborhood of a generic curve,and we got an open dense subset with the uniqueness result.Theorem1.1.Let A be the space of C3,µembeddings of S1into S2∞(H3).Then there exist open dense subset A′⊂A such that for anyΓ∈A′,there exist unique least area planeΣwith∂∞Σ=Γ.The organization of the paper as follows:In the next section we will give some preliminary results which will be used in the following sections.In section3,we will construct foliated neighborhood of a least area plane spanning a generic curve. By using this,we will show that there is a unique least area plane spanning that curve.In Section4,we will show that the curves with this property is indeed open dense.Finally,in the last section we will give some concluding remarks.1.1.Acknowledgements:I want to thank David Gabai for very useful conversa-tions,and helpful remarks.2.P RELIMINARIESIn this section,we will overview the basic results which we use in the following sections.Definition2.1.A minimal plane is a plane such that the mean curvature is0at every point.A least area plane is a plane such that any disk in the plane is least area among the disks with same boundary.A least area plane is minimal,but the converse is not true in general.Definition2.2.A linear operator between two Banach spaces is called Fredholm operator if the dimension of kernel and codimension of the image arefinite.The difference between these dimensions is called the Fredholm index of the operator.A map between two Banach manifolds is Fredholm map if the derivative is Fredholm operator at every point.A regular value is a point in the image such that for any point in the preimage(may be empty),the derivative is surjective.The classical theorem of the subject is the Sard-Smale theorem from[Sm].GENERIC UNIQUENESS OF LEAST AREA PLANES IN HYPERBOLIC SPACE3 Theorem2.1.(Sard-Smale)Let f:X→Y be a Fredholm map.Then the regular values of f are almost all of Y,i.e except a set offirst category.We will use the following spaces in the remaining part.Letµ>0.A={α∈C3,µ(S1,S2∞)|αembedding}D={u∈C3(S1,S1)|u diffeomorphism and satisfies three point condition, i.e.u(e23kπi,k=1,2,3}M={f:D2→H3|f(D2)minimal and f|∂D2∈A}Since A is an open submanifold of the space of immersions,we can replace it with the space of immersions in[Co1],and the same proofs will go through.So, we have the following theorem from[Co1].α◦u(D2)is minimal}is a submanifold of Theorem2.2.M={(α,u)∈A×D|the product bundle A×D,and the bundle projection map when restricted to M,π1|M:M→A,is Fredholm of index0.In other words,we realize M,the space of minimal planes,as a subspace of the product bundle A×D.Letα∈A,and consider thefiber({α}×D)∩M.α◦u i(D2),with asymptotic boundary Then we got the all minimal planes,Σi=Γ=α(S1),i.e∂∞Σi=Γ.So,the theorem says that the boundary restriction map (π1((α,u))=α)from the space of minimal maps to the the space of boundary parametrizations in above setting is Fredholm of index0.The next theorem which we use is the classical inverse function theorem for Banach manifolds,[La].Theorem2.3.(Inverse Function Theorem)Let M and N be Banach manifolds, and let f:M→N be a C p map.Let x0∈M and d f is isomorphism at x0.Then f is local C p diffeomorphism,i.e.there exist an open neighborhood of U⊂M of x0and an open neighborhood V⊂N of f(x0)such that f|U:U→V is C p diffeomorphism.Now,we can establish the starting point of the paper.Theorem2.4.Letα∈A be a generic curve as above.Then for anyΣ∈π−11(α),:UΣ→Vαis a there exist neighborhoods UΣ⊂M,and Vα⊂A such thatπ1|UΣhomeomorphism.Proof:By Theorem2.2,the mapπ1|M:M→A is Fredholm of index0.By Sard-Smale Theorem,the regular values are generic forπ1.Letα∈A be a regular value,andΣ∈π−11(α)⊂M.Sinceαis regular value,Dπ1(Σ):TΣM→TαA is surjective.Moreover,we know thatπ1is Fredholm of index0.This implies Dπ1 is isomorphism at the pointΣ∈M.By Inverse Function Theorem,there exist a neighborhood ofΣ,whichπ1maps homeomorphically onto a neighborhood of α∈A.4BARIS COSKUNUZERBeforefinishing this section,we want to quote another result from[Co1],which will be used in the following sections.This result is indeed true for C1curves by Hardt and Lin’s regularity result in[HL].Theorem2.5.LetΓ⊂S2∞(H3)be a C3,µJordan curve.Then,any least area plane spanningΓis properly embedded.3.F OLIATED N EIGHBORHOODS OF L EAST A REA P LANESIn this section,our aim is to construct a foliated neighborhood for any least area plane spanning a given generic curve(regular value of the Fredholm map)in A. Moreover,we will show that the leaves of this foliation are embedded least area planes whose asymptotic boundaries are disjoint from each other.By using this, we will show that uniqueness of the least area plane spanning the regular curve. We will abuse the notation by using interchangeably the mapΓ:S1→S2∞(H3) with its imageΓ(S1).Similarly same is true forΣ:D2→H3and its image Σ(D2).LetΓ0∈A be a generic curve,and letΣ0∈π−11(Γ0)⊂M be a least area plane whose existence guaranteed by Anderson’s result in[A1].Then by Theorem 2.4,there is a neighborhood ofΣ0∈U⊂M homeomorphic to the neighborhood Γ0∈V⊂A.LetΓ:[−ǫ,ǫ]→V be a path such thatΓ(0)=Γ0and for any t,t′∈[−ǫ,ǫ],Γt∩Γt′=∅.In other words,{Γt}foliates a neighborhood ofΓ0in S2∞(H3).Let Σt∈U be the preimage ofΓt under the homeomorphism.Lemma3.1.{Σt}is a foliation of a neighborhood ofΣ0in H3by embedded least area planes.Proof:We will prove the lemma in three steps.Claim1:For any s∈[−ǫ,ǫ],Σs is an embedded plane.Proof:SinceΣ0is a least area plane,by Theorem2.5,Σ0is an embedded plane.Now,{Σt}is continuous family of minimal planes.We cannot apply the Theorem2.5to these planes,since the theorem is true for least area planes,while our planes are only minimal.Let s0=inf{s∈(0,ǫ]|Σs is not embedded}.But,since{Σt}is continuous family of planes,and this can only happen whenΣshas tangential self intersection (locally lying on on side).But this contradicts to maximum principle for minimal surfaces.So for all s∈[0,ǫ],Σs is embedded.Similarly,this is true for s∈[−ǫ,0], and the result follows.Claim2:{Σt}is a foliation,i.e.for any t,t′∈[ǫ,ǫ],Σt∩Σt′=∅.Proof:Assume on the contrary that there exist t1<t2such thatΣt1∩Σt2=∅.First,since the asymptotic boundariesΓt1andΓt2are disjoint,the intersectionGENERIC UNIQUENESS OF LEAST AREA PLANES IN HYPERBOLIC SPACE5 cannot contain an infinite line.So the intersection must be a collection of closed curves.We will show that in this situation,there must be a tangential intersection between two planes,and this will contradict to the maximum principle for minimal surfaces.IfΣt2does not intersect all the minimal planesΣs for s∈[−ǫ,t2],let s0= sup{s∈[−ǫ,t2]|Σt2∩Σs=∅}.Then,since{Σt}is continuous family ofminimal planes,it is clear thatΣt2must intersectΣstangentially,and lie in oneside ofΣs.But this contradicts to maximum principle for minimal surfaces.So,let’s assumeΣt2intersects all minimal planesΣs for s∈[−ǫ,t2].Let s0=sup{s∈[−ǫ,t2]|Σ−ǫ∩Σs=∅}.SinceΣ−ǫis embedded by Claim1, there exist an s0∈(−ǫ,t2].But again since{Σt}is continuous family of minimal planes,Σs must intersectΣ−ǫtangentially,and lie in one side ofΣs.But this again contradicts to maximum principle for minimal surfaces.Claim3:For any s∈[−ǫ,ǫ],Σs is a least area plane.Proof:FixΣs for s∈(−ǫ,ǫ).Now,let[Σ−ǫ,Σǫ]be the region bounded by embedded planesΣ−ǫandΣǫin H3.By above results,Σs⊂[Σ−ǫ,Σǫ].Since the boundaries are least area planes,[Σ−ǫ,Σǫ]is a mean convex region.Letγ⊂Σs be a simple closed curve.By Meeks and Yau’s theorem[MY],there exist a least area embedded disk D spanningγin the mean convex domain[Σ−ǫ,Σǫ].If D is not inΣs,it must intersect other leaves nontrivially.Then{Σt}∩D induce a singular 1-dimensional foliation F on D.The singularities of the foliation are isolated as {Σt}are minimal planes.Since Euler characteristic of the disk is1,by Poincare-Hopf index formula there must be a positive index singularity implying tangential (lying on one side)intersection of D with some leaveΣs.But this contradicts to maximum principle for minimal surfaces.Sinceǫwas chosen arbitrarily at the beginning,one can start with suitableǫ′>ǫ.The whole proof will go through,and this shows thatΣ±ǫare also least area.The result follows.Lemma3.2.Σ0is the unique least area plane with asymptotic boundaryΓ0.Proof:LetΣ′be another least area plane with asymptotic boundaryΓ0.If Σ0=Σ′thenΣ′must intersect a leave in the foliated neighborhood ofΣ0,sayΣs. But,since∂∞Σs=Γs is disjoint from∂∞Σ′=Γ0,this implies the intersection cannot have infinite lines but closed loops.But these are least area planes,and by Meeks-Yau’s exchange roundoff trick,two least area plane cannot intersect in a closed loop.This is a contradiction.So,we have proved the following theorem:6BARIS COSKUNUZERTheorem3.3.LetΓ∈A be a generic curve as described above.Then there exist a unique least area planeΣ⊂H3with∂∞Σ=Γ.Remark3.1.This theorem does not say that there exist a unique minimal plane spanning a given generic curve.In our proof(Lemma3.2),we essentially use the plane being least area.So far we have proved the uniqueness of least area planes for a subsetˆA⊂A which is except afirst category set.In the next section,we will show that this is true for a more general class of curves,i.e.an open dense subset of A.4.O PEN DENSE SET OF CURVESIn this section,we will show that any regular curve has an open neighborhood such that the uniqueness result holds for any curve in this neighborhood.LetΓ0∈A be a regular curve,and letΣ0∈π−11(Γ0)⊂M be the unique least area plane spanningΓ0.Let U⊂M be the neighborhood ofΣ0homeomorphic to the neighborhood V⊂A ofΓ0as above.We will show thatΓ0has a smaller open neighborhood V′⊂V such that for anyΓ∈V′,there exist unique least area plane in H3with∂∞Σ=Γ.First we will show that the curves disjoint fromΓ0in the open neighborhood also bounds a unique least area plane in H3.Lemma4.1.Letβ∈V withβ∩Γ0=∅.Then there exist a unique least area plane spanningβ.Proof:Sinceβ∈V is disjoint fromΓ0,we canfind a pathΓ:(−ǫ,ǫ)→V, such that{Γt}foliates a neighborhood ofΓ0in S2∞(H3),andβis one of the leaves, i.e.β=Γs for some s∈(−ǫ,ǫ).Then the proofs of the previous section implies thatΣβ=Σs and{Σt}also gives a foliation of a neighborhood ofΣβby least area planes.Then proof of Lemma3.2implies thatΣβis the unique least area plane spanningβ.Now,if we can show same result for the curves in V intersectingΓ0,then we are done.Unfortunately,we cannot do that,but we will bypass this by going to a smaller neighborhood.Lemma4.2.There exist a neighborhood V′⊂V ofΓ0such that for anyΓ′0∈V′, there exist a unique least area plane with asymptotic boundaryΓ′0.Proof:Let V′⊂V be an open neighborhood containingΓ0such that for anyΓ′0∈V′,there exist two curvesβ1,β2∈V,such thatβ1andβ2are both disjoint fromΓ0andΓ′0.We also assume that if B⊂S2∞(H3)is the annulus bounded byβ1andβ2,Γ0,Γ′0⊂B.To see the existence of such a neighborhood, one canfix two curves disjoint fromΓ0,and lying in the opposite sides ofΓ0inGENERIC UNIQUENESS OF LEAST AREA PLANES IN HYPERBOLIC SPACE7 S2∞(H3).Then suitable complements of these two curves in V will give us the desired neighborhood ofΓ0.Now,fixΓ′0∈V′.By the assumption on V′,there are two curvesβ1,β2disjoint from bothΓ0,Γ′0and bounding the annulus B in S2∞(H3)such thatΓ0,Γ′0⊂B⊂S2∞(H3).Then,we canfind two pathsΓ,Γ′:[−ǫ,ǫ]→V with{Γt},{Γ′t}foliates B such thatΓ(ǫ)=Γ′(ǫ)=β1,Γ(−ǫ)=Γ′(−ǫ)=β2,andΓ(0)=Γ0,Γ′(0)=Γ′0.By Lemma4.1,we know that{Γt}induces{Σt}family of embedded least area planes with asymptotic boundary{Γt}.Moreover,these least area planes are unique with the given asymptotic boundary,and leaves of the foliation in the neigh-borhood ofΣ0.Now,consider the preimage of the pathΓ′under the homeomorphismπ1|U: U→V.This will give us a pathΣ′⊂U⊂M,which is a continuous family of minimal planes,say{Σ′t}.We claim that this is also a family of embedded least area planes inducing a foliated neighborhood ofΣ′0.By previous paragraphs we know thatΣǫand theΣ−ǫare the unique least area planes with asymptotic boundaryβ1andβ2,respectively.This meansΣ′±ǫ=Σ±ǫ.So,the family{Σ′t} has embedded least area planesΣ′±ǫ.Then by slight modification of the proof of Lemma3.1shows that{Σ′t}is a family of embedded least area planes inducing a foliation of a neighborhood ofΣ′0.By Lemma3.2,Σ′0is the unique least area plane spanningΓ′0.So,we got the following theorem.Theorem4.3.There exist an open dense subset A′⊂A,where A={α∈C3,µ(S1,S2∞)|αembedding}with C3,µtopology,such that for anyΓ∈A′,there exist unique least area plane with asymptotic boundaryΓ.Proof:The set of regular values of Fredholm map,sayˆA,is the whole set ex-cept a set offirst category by Sard-Smale theorem.So,the regular curves are dense in A.By above lemmas,for any regular curveΓ0,there exist an open neighborhoodV′Γ0⊂A which the uniqueness result holds.So,A′= Γ∈ˆA V′Γis an open densesubset with the desired properties.5.F INAL R EMARKS5.1.The technique.The technique we used in this paper to show uniqueness of the least area planes is similar to the techniques of Anderson and Lin for their proof for convex domains[A1]and star shaped domains[Li1]in S2∞(H3).The idea is simple.Fix the least area plane.Construct a neighborhood foliated by least area planes with disjoint asymptotic boundaries.Then any other least area plane with same asymptotic boundary with the original one,must intersect the other least area planes.But this intersection cannot contain an infinite line as the asymptotic8BARIS COSKUNUZERboundaries are disjoint.So the intersection must be a collection of closed curves. But Meeks-Yau’s exchange roundoff trick says that two least area planes cannot intersect in a closed loop,and gives the desired contradiction.On the other hand,hyperbolic space and its asymptotic boundary are not very essential for this technique.One can employ same method for any convex domain, and least area disks whose boundaries are in the boundary of this convex domain, once one has the continuous family of minimal disks.5.2.Extremal Curves in R3.As we mentioned above,we can use this technique for extremal curves in convex domains in R3.Let N⊂R3be a smooth convex domain.Then by changing the space of curves from embeddings of S1into R3 to the embeddings of S1into∂N in[TT],the whole proof would go through.So this will establish the necessary analytical background to employ the techniques in this paper.Then one canfix a regular extremal curve in∂N,and get a foliated neighborhood of the embedded least area disk spanning thefixed curve in N.This proves again uniqueness of least area disk.So,one can show that for a given smooth convex domain N in R3,there exist open dense subset of C3,µJordan curves in∂N bounds a unique least area disk in R3.In some sense,this is generic uniqueness for extremal curves in R3.By using the same setting and similar techniques,Fang-Hua Lin showed that an extremal curve either bounds a unique least area disk,or two”extremal”stable minimal disk in[Li2].One can get same result by using the methods of this paper. Also,by using similar techniques in an analytical way,Li-Jost proved that a C3,µJordan curve in H3with total curvature less than4πbounds a unique least area disk,[Lj].There is a great resemblance between asymptotic Plateau problem in hyperbolic space,and Plateau problem for the extremal curves in Euclidean space from geo-metric point of view.We believe that most of the results for extremal curves can be carried out to the asymptotic Plateau problem case.5.3.Questions.As we mentioned in the introduction,there are not many results on the number of solutions of asymptotic Plateau problem in H3.In this paper,we showed a generic uniqueness result for some smooth class of curves.One suspects whether there is any smooth class of Jordan curves in S2∞(H3)such that any curve in this class bounds onlyfinitely many least area planes.Another question in the opposite direction is whether there is any smooth Jordan curve in S2∞(H3)bounding infinitely many least area planes.If you remove the condition being plane(the topological type of the surface is disk),Anderson gave examples of Jordan curves bounding infinitely many area minimizing surfaces in[A2].If one remove the embeddedness condition for the curve,the universal cover of a hyperbolic manifoldfibering over a circle,induces aGENERIC UNIQUENESS OF LEAST AREA PLANES IN HYPERBOLIC SPACE9 Peano curve in S2∞(H3)bounding infinitely many minimal planes,corresponding to the universal cover of the least area representative of thefiber.It is still an open question tofind a Jordan curve bounding infinitely many least area planes.But it is reasonable to hope that the bridge principle hold in this case too,and one constructs a rectifiable embedded curve in S2∞(H3)similar to the Euclidean case.R EFERENCES[A1]M.Anderson,Complete minimal varieties in hyperbolic space,Invent.Math.69,477–494 (1982).[A2]M.Anderson,Complete minimal hypersurfaces in hyperbolic n-manifolds,Commnet.Math.Helv.58,264–290(1983).[Co1] B.Coskunuzer,Minimal Planes in Hyperbolic Space,Comm.Anal.Geom.12,821–836 (2004).[Co2] B.Coskunuzer,Uniform1-cochains and Genuine Laminations,eprint;math.GT/0304094 [G] D.Gabai,On the geometric and topological rigidity of hyperbolic3-manifolds,J.Amer.Math.Soc.10,37–74(1997).[HL]R.Hardt and F.H.Lin,Regularity at infinity for area minimizing hypersurfaces in hyperbolic space,Invent.Math.88,217–224(1987).[HS]J.Hass and P.Scott,The Existence of Least Area Surfaces in3-manifolds,Trans.AMS310, 87–114(1988).[L]ng,Asymptotic Plateau problem in Gromov hyperbolic manifolds,Calc.Var.Partial Differential Equations16,31–46(2003).[La]ng,Real analysis,Addison-Wesley,MA,(1983).[Li1] F.H.Lin,On the Dirichlet problem for the minimal graphs in hyperbolic space,Invent.Math.96,593–612(1989).[Li2] F.H.Lin Plateau’s problem for H-convex curves,Manuscripta Math.58,497–511(1987). [Lj]X.Li-Jost Uniqueness of minimal surfaces in Euclidean and hyperbolic3-space,Math.Z.217,275–285(1994).[LT1]P.Li and L.F.Tam,Uniqueness and regularity of proper harmonic maps,Ann.of Math.(2),137,167–201(1993).[LT2]P.Li and L.F.Tam,Uniqueness and regularity of proper harmonic maps.II,Indiana Univ.Math.J.42,591–635(1993).[MY]W.Meeks and S.T.Yau,The classical Plateau problem and the topology of three manifolds, Topology21,409–442(1982)[Sm]S.Smale,An infinite dimensional version of Sard’s Theorem,Amer.J.Math.87,861–866 (1965).[Tr] A.J.Tromba,On the number of simply connected minimal surfaces spanning a curve,Mem.Amer.Math.Soc.,no.194(1977).[TT] F.Tomi and A.J.Tromba,Extreme curves bound embedded minimal surfaces of the type of the disc,Math.Z.158,137–145(1978).D EPARTMENT OF M ATHEMATICS,Y ALE U NIVERSITY,N EW H AVEN,CT06520E-mail address:baris.coskunuzer@。