On the Hofer Girth of the Sphere of Great Circles
aa r X i v : . [ m a t h . S G ] O c t ON THE HOFER GIRTH OF THE SPHERE OF GREATCIRCLES
ITAMAR ROSENFELD RAUCH
Abstract.
An oriented equator of S is the image of an oriented embed-ding S ֒ → S such that it divides S into two equal area halves. FollowingChekanov, we define the Hofer distance between two oriented equators as theinfimal Hofer norm of a Hamiltonian diffeomorphism taking one to another.Consider E q + the space of oriented equators. We define the Hofer girth of anembedding j : S ֒ → E q + as the infimum of the Hofer diameter of j ′ ( S ), where j ′ is homotopic to j . There is a natural embedding i : S ֒ → E q + , sending apoint on the sphere to the positively oriented great circle perpendicular to it.In this paper we provide an upper bound on the Hofer girth of i . Introduction
Definitions and main result.
Let S ⊂ R be the two dimensional unitsphere, equipped with a symplectic structure ω = π ω std such that Area( S ) = 1.An (oriented) equator L of S is the image of an (oriented) embedding S → S such that it divides the sphere into two equal area halves. Denote by E q the spaceof unoriented equators, and likewise by E q + the space of oriented equators. Thispaper is a part of an endeavor to better understand the geometry of the S equatorsspace. While not a direct sequel to [Kha09], and [Kha15], this work is inspired bythese articles in an attempt to investigate special Lagrangians on surfaces.Consider a compact Riemannian manifold ( X, g ). Recall that the systole of X is a metric invariant of X , defined as the least length of a noncontractible loop in X . This sort of 1-dimensional question is classical in geometry. Higher dimensionalgeneralizations go back to Loewner, Berger, Pu, and others; see [Gro96] for a survey.The work presented here may be considered as one way to generalize systoles tospheres in E q + . Roughly speaking, we would like to find the least Hofer diameterof an embedded noncontractible sphere in E q + .To define this Hofer diameter we need to recall the definition of Hofer distance;for more details see, e.g., [MS17]. Let ( M, ω ) be a connected symplectic manifoldwithout boundary. Denote by Ham c ( M ) the group of compactly supported Hamil-tonian symplectomorphisms. The Hofer distance between φ , φ ∈ Ham c ( M ) isgiven by ρ ( φ , φ ) = inf φ H = φ ◦ φ − Z max p ∈ M H ( t, p ) − min p ∈ M H ( t, p ) dt, where the infimum goes over all time dependent compactly supported smoothHamiltonians H : [0 , × M → R that generate φ ◦ φ − . The author was partially supported by the Azrieli Foundation.
In this paper we only consider S as our symplectic manifold, and since it iscompact Ham c ( S ) = Ham( S ). Consider the following distance function on E q + ,introduced by Chekanov in [Che00]. Definition.
Let L , L ⊂ S be two oriented equators, define their distance by d H ( L , L ) := inf φ ∈ Ham( S ) φ ( L )= L k φ k . The Hofer distance induces the following definitions of diameter and girth.
Definition.
Let f : S → E q + . Put, (1) diam( f ) := sup x,y ∈ S d H ( f ( x ) , f ( y )) . Let α ∈ π ( E q + ) . Define, (2) girth( α ) = inf f ∈ α diam( f ) . Our main result is the following.
Theorem 1.
Let i : S → E q + denote the embedding sending each point on thesphere to the positively oriented great circle perpendicular to it. Then, the followingbound holds for the homotopy class [ i ] ∈ π ( E q + ) : girth([ i ]) ≤ / . Note that the Hofer diameter of i is 1 /
2, as we explain in Subsection 1.2. Theproof of Theorem 1 consists in assembling the pieces from the following sections,for completeness we provide it here.
Proof.
Fix some sufficiently small δ, ǫ >
0. We construct in Section 2 a homotopydepending on δ , and ǫ , between i and some embedding i p , which we call a “pipeequator embedding”. We calculate in Section 3 that 1 / δ is an upper bound forthe Hofer diameter of i p . Since δ is arbitrary, Theorem 1 follows. (cid:3) Remark that there are other interesting attempts at understanding the geometryof E q + . We would like to note one such attempt by Y. Savelyev, somewhat close inspirit to the contents of this paper; See [LS14] and Savelyev’s preprint in [Sav15] ,and more advanced versions of this paper in the author’s website. Savelyev cal-culates a similar notion to the girth above in the space of loops of equators withrespect to the positive Hofer length functional.1.2. Four elementary observations.
First, the homotopy class of i is non-trivial; see Section 4 for details. Hence, whether or not girth([ i ]) is zero is a non-trivial question. Second, we have the following naive upper bound on girth([ i ]),(3) diam( i ) = 12 . We claim that the Hofer distance between any two great circles is bounded aboveby 1 /
2. Indeed, recall that SO (3) is a subgroup of Ham( S ); see, e.g. Section 1.4 in[Pol01]. Thus the Hofer distance between any two great circles is bounded above bythe Hofer norm of the rotation taking one to another. Under the choice of symplectic We thank Egor Shelukhin for referring us to this paper.
N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 3 form with Area( S ) = 1, it holds that the Hofer norm of such rotations is boundedabove by 1 /
2. To see this, let R = R ( θ, u ) be a counterclockwise rotation with angle θ about the axis prescribed by the unit vector u . It is straightforward to checkthat the height function with respect to u , i.e., F ( x ) = ( x, u ) defined for x ∈ S ,where here ( · , · ) stands for the standard Euclidean inner product, generates R as aHamiltonian diffeomorphism; see example 1.4.H in [Pol01]. The time parameter ofthe flow generated by F determines the angle θ for R . Given an angle θ ∈ [0 , π ], itfollows that k R ( θ, u ) k ≤ Z θ max x ∈ S F ( x ) − min x ∈ S F ( x ) dt = 2 θ. Rescaling by 4 π to take our choice of area form into account, yields k R ( θ, u ) k ≤ θ π ≤ . This shows the upper bound in (3); and in particular,girth([ i ]) ≤ . The lower bound in (3) follows from the well-known energy-capacity inequality,which takes the following simpler form for surfaces. Given a closed symplecticsurface (
M, ω ), for any connected subset A ⊂ M it holds that e ( A ) ≥ Area( A ).To apply it to our case, note that i ( x ) bounds a disc D ⊂ S of area 1 /
2, and that i ( − x ) is the same great circle as i ( x ) with orientation reversed. Meaning, anyHamiltonian diffeomorphism φ taking i ( x ) to i ( − x ) would displace D , and so wefind that k φ k ≥ e ( D ) ≥ Area( D ) = 12 . Hence, Equation (3) holds.The third elementary observation is that if we consider the space of unorientedequators E q , and the embedding ι : R P → E q induced by i , we find that,girth([ ι ]) = diam( ι ) = 1 / . By this we mean that the Hofer diameter of ι is 1 /
4, and maps homotopic to ι haveHofer diameter of at least 1 /
4. To define ι precisely, let q ∈ R P ; lift it to ˜ q ∈ S viathe double cover S → R P . Then, put ι ( q ) to be the great circle i (˜ q ) without theorientation. Due to the composition with the orientation forgetful map E q + → E q ,it holds that ι ( q ) does not depend on the lift ˜ q .We claim now that d := diam( ι ) = 1 /
4. For ease of notation, consider theinduced Hofer distance on R P , defined by d H ( x, y ) = d H ( ι ( x ) , ι ( y )) , ∀ x, y ∈ R P . So, d is the diameter of R P under this distance, i.e., d = diam( R P ) := sup x,y ∈ R P d H ( x, y ) . To prove the claim, observe that an upper bound for d is obtained by a rotation by π/
2, similar to the case of the second elementary observation above. For the lower Recall that the displacement energy e ( A ) for a connected A ⊂ ( M, ω ) is defined to be inf k φ k ,where the infimum goes over φ ∈ Ham c ( M ) with A ∩ φ ( A ) = ∅ . ITAMAR ROSENFELD RAUCH bound, note that for each point q ∈ R P there exists another point p ∈ R P , suchthat d H ( q, p ) ≥ / . To see this, consider any non-contractible loop in R P based in q . It lifts to acurve γ in S connecting two antipodal points x, − x , where q is the class of { x, − x } under the covering map S → R P . By (3), it follows that d H ( i ( x ) , i ( − x )) ≥ / γ is at least 1 /
2. Moreover, there exists a point ˜ p ∈ S lying on γ , such that d H ( i (˜ p ) , i ( x )) ≥ /
4, and likewise d H ( i (˜ p ) , i ( − x )) ≥ / p to R P yields a point p ∈ R P with d H ( q, p ) ≥ / . Let H : R P × I → E q be a homotopy of embeddings R P → E q , where I =[0 ,
1] and H = ι . The arguments above may be repeated verbatim to show thatdiam( H t ) ≥ / t ∈ I . Indeed, H may be lifted to a homotopy ˜ H : S × I →E q + such that ˜ H = i . Let q ∈ R P , and γ a non-trivial loop based at q as before.Again, γ lifts to a path in S connecting x to − x , where q is the class of { x, − x } .Note that the following diagram commutes S × I E q + R P × I E q ˜ H prj × fH where, prj : S → R P is the covering map, and f : E q + → E q is the orientationforgetful map. Hence, ˜ H t ( x ) and ˜ H t ( − x ), are the same unoriented equator, forall t ∈ I . Since i ( x ) and i ( − x ) have opposite orientations, so does ˜ H t ( x ) and˜ H t ( − x ) for all t ∈ I , by continuity. Now, for any t ∈ I , the arguments beforemay be repeated to find a point ˜ y ∈ S , such that d H ( ˜ H t (˜ y ) , ˜ H t ( x )) ≥ /
4, and d H ( ˜ H t (˜ y ) , ˜ H t ( − x )) ≥ /
4. Put y = prj(˜ y ) the projection of ˜ y to R P . We havethus found that, d H ( H t ( y ) , H t ( q )) ≥ . Therefore, diam( H t ) ≥ / t ∈ I , as claimed.The fourth and final elementary observation is that the naive bound of 1 / i has a strictly smaller diameter.The idea of the construction is to perturb i such that no equator in its imageappears together with its oppositely oriented copy. To bound the Hofer distancebetween perturbed equators we use the following lemma. Lemma 1.
Let
L, L ′ ⊂ S be two equators obtained as C -small perturbations of agreat circle, taken with opposite orientations, such that L ∩ L ′ < ∞ . Then, d H ( L, L ′ ) < . Before proving Lemma 1, let us first go into some of the details of the construc-tion. Observe that in a small neighborhood U of a great circle L , one may applya Hamiltonian diffeomorphism to perturb L to a graph of a smooth function ofthe angle parameter of L . Indeed, let ( q, p ) be coordinates in U , such that q isa parameter along L , and p a coordinate in an orthogonal direction. Consider S as R / π Z , and let f ( q ) : S → ( − δ, δ ) be a continuous function with f (0) = 0, and N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 5 R S f ( t ) dt = 0, where δ > f is contained in U when con-sidered as a function along L . Then, H ( q, p ) = R q f ( t ) dt is a Hamiltonian, whosetime 1 flow, φ H , satisfies that φ H ( L ) is the graph of f .Choose an open cover of S by disjoint pairs of antipodal topological discs, de-noted by U = { U , V , . . . , U k , V k } , where U j and V j are antipodal. For ease ofnotation, put L p = i ( p ) for p ∈ S . In each neighborhood W ∈ U , pick aparametrization for all the great circles in i ( W ). Pick distinct prime numbers r j , s j for each pair of U j , V j ∈ U , and use the above parametrizations to define,for each j = 1 , . . . , k , and each p ∈ U j , q ∈ V j , functions f p : L p → ( − δ, δ ), g q : L q → ( − δ, δ ), by f p ( t ) = δ sin( r j t ), g q ( t ) = δ sin( s j t ). Using a partition ofunity subordinate to U , we merge these functions to ˆ f p : L p → ( − δ, δ ), defined forall p ∈ S , such that it is C -small. By construction, for all p ∈ S it holds thatthe graphs of ˆ f p , and ˆ f − p are C -small perturbation of the same unoriented greatcircle, with a finite number of intersection points. Thus, Lemma 1 applies.By the second elementary observation above, the Hofer diameter of i is realizedby antipodal points. The above perturbation results in an embedding i : S → E q + ,where each pair of points are at Hofer distance strictly less than 1 / i ]) < / , as claimed. Idea of the Proof of Lemma 1.
Denote by D , D the filling discs of L , L ′ respec-tively, such that D ∩ D has area ǫ ≪ /
2. Pick a point p in S outside of D ∪ D ,and “puncture” the sphere at p to obtain the following picture in R ; see Figure1. The two discs, D and D , project to two regions in R , depicted in Figure 1 D D D D Figure 1.
Perturbing equators to create an embedding with di-ameter smaller than 1 / D and D as well. Note that each of these has area 1 /
2. By the hypothesis, theintersection D ∩ D has finitely many connected components; let F be the one oflargest area. Denote by ǫ ′ the area of F , so the area of D ∩ D \ F is δ = ǫ − ǫ ′ .Using techniques as in [Kha15], which we demonstrate more thoroughly in Section3, flow the smaller connected components into F , through D , such that their areais “pushed” into D . A sketch of the flow is demonstrated in the Figure 2, wherethere are only two connected components for the intersection. Of course, this is ITAMAR ROSENFELD RAUCH D D ′ Figure 2.
Flowing the intersection’s connected components intothe largest component through a flow described intuitively here byarrows.possible since D has area 1 /
2, whereas the area flowed into D is (slightly largerthan) δ , which is by assumption much smaller than 1 /
2. Remark that an “excess”area is flowed for technical reasons which we omit here. Since D ∩ D has a fi-nite number of components, we may apply this procedure repeatedly until a singleconnected component remains.Denote by D ′ , and D ′ , the images of D , and D , respectively, under theprocedure above. Observe the following few properties. First, the intersection F ′ := D ′ ∩ D ′ has a single connected component of area ǫ . Second, both D ′ \ F ′ and D ′ \ F ′ have area 1 / − ǫ . Third and last, since L , L ′ , are graphs with respect to thesame great circle, the resulting D ′ and D ′ each have only one connected component.Hence, after a change of coordinates we may consider D ′ , and D ′ , as two discs withthe above areas and an intersection of area ǫ . Since Area( D ′ \ F ′ ) = Area( D ′ \ F ′ ),it follows that, d H ( L, L ′ ) ≤ d H ( ∂D ′ , ∂D ′ ) + δ = 12 − ǫ + δ = 12 − ǫ ′ < , as claimed. (cid:3) First look at pipe equators.
In this subsection we would like to introducein rather loose terms the concept of “pipe equators”. Let us begin with consideringa cartoon of a typical equator we wish to examine, in Figure 3, and explain themotivation for the concrete construction which will follow in Sections 2 and 3. Assomewhat of a static picture, the red curve in Figure 3 may be obtained by gluingtwo equal area slit discs to one another to form a sphere. Choosing the discs to havearea 1 / δ each, results in a curve on the gluedsphere, which separates it to two equal area connected components. Therefore, thiscurve is in fact an orientable equator. For the rest of the paper, these cut out stripswill be called “pipes”, and equators formed this way will be called “pipe equators”.Observe that these pipe equators are determined by three parameters: the area δ of the pipes (which we think of as fixed); the area a to one fixed side of the upperpipe; and b the area to one fixed side of the lower pipe. Remark that we choseto begin with this static picture in order to more easily explain the obstruction tothe parameters determining pipe equators, which in turn leads to the homotopyconstructed in Section 2. In practice, a more useful approach to these equators isdynamical in essence, and is investigated further in Section 3. N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 7
Figure 3.
Pipe equator in red; the blue region denotes a con-nected component of the sphere with the equator removed.Let us briefly discuss these parameters and consider an unsuccessful yet in-structive first attempt at constructing an embedding S → E q + via pipe equators.Observe that a priori a ∈ (0 , / a = 0 , / L and so a pipe equator is undefinable. Similarly, b ∈ (0 , / I = (0 , / I as the pipe equators’ parameters space. The above suggests theexistence of an embedding ι : I → E q + , such that the boundary of I should bemapped to L . Thus, one might naively attempt to collapse ∂I to a point in orderto obtain a map S → E q + ; however, this map remains ill defined. Since the areaon the other side of the pipe in the upper hemisphere is 1 / − a − δ , and like a it is bounded in (0 , / a cannot be more that 1 / − δ ; a similarargument shows that b should lie in (0 , / − δ ).To properly define such a continuous embedding, we homotope between i andan embedding i p : S → E q + with the following property; namely, it is obtainedas a continuous transition from a constant map near the boundary ∂I , to ι awayfrom it; see Section 2 for details about this construction. Of course, i p belongs inthe same homotopy class of i in π ( E q + ). We show below that its Hofer diameteris bounded above by 1 / δ . Essentially, this is attained by equators representedby points far from the boundary ∂I , i.e., by pipe equators. Structure of the paper.
Section 2 constructs a homotopy between the natural em-bedding i and an embedding whose image mostly consists of pipe equators. Then,Section 3 shows that such an embedding has a Hofer diameter of 1 /
3, up to anarbitrarily small parameter. For completeness, Section 4 shows that π ( E q + ) = 0and that the homotopy class of i is non-trivial. Acknowledgements.
I would like to thank Dr. M. Khanevsky for his contributionto this project, as well as for his patience and guidance.2.
Homotopy between embeddings
In this section we construct a homotopy between i , and an embedding S ֒ → E q + which depends on 0 < δ ≪ ITAMAR ROSENFELD RAUCH construction details, first fix some 0 < ǫ, δ ≪
1, where δ denotes the area of apipe in a pipe equator as in Subsection 1.3, and ǫ will be used below to demarcatea region of the equators’ parameter space where the homotopy transitions fromthe constant map to a pipe equator embedding. Let us now introduce a suitableparametrization of the sphere. It will be used throughout this section, startingwith defining the projection to the unit disc. Remark that in order to describe thisparametrization slightly more easily, we reverse in this section the orientations ofequators in the image of i . This reversal does not interfere with the generality ofour arguments.Denote by q ∈ S the south pole of the sphere. Observe that every point on S \ { q } lies on a circle through q whose angle with respect to the xz -plane is φ ∈ ( − π/ , π/ θ ∈ (0 , π ), suchthat θ = 0 , π refers to q ; see Figure 4. Denote these circles by C φ ; note that theyare obtained as the intersection of S with affine planes Π φ forming an angle φ withthe xz -plane and parallel to the x -axis. θφ φθ θ = 0 φ = − π π π q x yz Figure 4.
Parametrization of the sphere by angles.Let us construct a projection to the planar unit disc. Fix some 0 ≤ φ < π/ − ǫ ,and consider the equators given by i ( x ), where x ∈ C φ . These are rotations of L about an axis, l φ , perpendicular to Π φ such that it passes through the centerof S , and its positive direction is determined by the intersection with the upperhemisphere. Indeed, each x ∈ C φ determines a counterclockwise rotation R = R ( θ, l φ ) of S , with angle θ about the axis l φ , where θ = θ ( x ) is the θ coordinateof x . Observe that as L rotates, it forms an open cap in the upper hemisphere, D , consisting of points which are unswept by the rotated L . Note that D is thespherical cap formed about the axis l φ in the positive direction, with an angle φ measured at the center of S , relative to l φ . Denote the center of D by x φ . As acounterpart to D , its reflection with respect to Π φ is a cap D contained in the lowerhemisphere, H , and tangent to L , so it consists of points that are always containedin H . Indeed, as x ranges over C φ , the lower hemisphere, H , and D , rotate via N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 9 R ( θ, l φ ), for θ = θ ( x ). Clearly, R ( θ, l φ )( D ) is contained in H θ = R ( θ, l φ )( H ) andis tangent to L θ = R ( θ, l φ )( L ) for all θ ∈ (0 , π ).Puncture the sphere at x φ to obtain a projection to the unit disc, i.e., a diffeo-morphism f : S \ { x φ } → D to the open disc of area 1, such that it preserves areasand translates rotations about l φ to rotations about the center of the disc. Notethat the inverse of this diffeomorphism can be described by taking the quotient ofthe open unit disc by its boundary, thus collapsing it to a point. The picture in thedisc as appears in Figure 5. f ( L ) D f ( ∂ D ) f ( D ) D f S Figure 5.
Projection of the sphere to the unit disc with somedistinguished regions.Remark that the rotation translation requirement on f has the following inter-pretation in Figure 5. As x ranges over C φ the rotation it induces on H translatesunder f to a rotation of the light blue region in the disc such that it always containsthe dark disc, tangent to it and to the red circle.Denote by H ⊂ D the region f ( H ), it has half the area of the disc. Considernow the following geometric homotopy, associated with θ = 0 , π , defined as inFigure 6, and denoted by F ′ t . F ′ ( H ) = H P Figure 6.
Homotopy of the disc.That is, the homotopy is defined by deforming H , starting at the picture on theleft to that on the right, such that it preserves areas; particularly, Area( F ′ t ( H ))is independent of t . On the left H remains unchanged as F ′ is the identity map.On the right we have F ′ ( H ) contained inside a slit annulus. Denote the slit by P and the inner disc of the annulus by S , as in the figure. The homotopy F ′ may be constructed such that Area( P ) = δ , andArea( S ) = 1 − δ ǫ − π φ − δ . Particularly, Area( S ) = 0 for φ = π/ − ǫ , and Area( S ) = 1 / − δ/ φ = 0.To define F ′ t for θ ∈ (0 , π ), put F ′ t,θ = R θ ◦ F ′ t ◦ R − θ , t ∈ [0 , , θ ∈ [0 , π ] , where R α is a positive rotation of the disc in an angle α with respect to the x -axisin the usual planar coordinates. The upshot is that when θ goes over (0 , π ) theslit P is rotated within the annulus. To make use of this behavior, we would like toapply a change of coordinates which translates the rotation in the annulus to a flowof the lower hemisphere through a pipe. Namely, define the change of coordinates g : D → D as in Figure 7, such that g ( S ) coincides with f ( S ), where S ⊂ S isthe area bounded by the pipe, P = f − ( P ), in the upper hemisphere, as appearsin Figure 8. We further require g to preserve areas. P g D D Figure 7.
Change of coordinates in the disc leading to a projec-tion of a pipe connected to a hemisphere. P Figure 8.
Hemisphere with a pipe attached to it.Then, as θ varies, we obtain the following picture after applying g ; see Figure 9. N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 11
Figure 9.
Rotation followed by a change of coordinates in the disc.By the well known Alexander trick, it holds that g is homotopic to the identity,since it is compactly supported. Let g t be a homotopy, such that g is the identity, g = g , and for all t ∈ [0 , g t preserves areas. Note that g t can bechosen such that Area( g t ( S )) is constant with respect to t . Define F t,θ = g t ◦ F ′ t,θ , Note that F t,θ preserves the area of H as t and θ vary, so the boundary ∂ [ f − ( F t,θ ( H ))]is an equator of S . The equator obtained by this procedure for t = 1 appears inFigure 10 as the bold black outline of the blue region (the dashed sections are inthe back of the sphere). This equator is precisely a pipe equator, as constructed inSubsection 1.3. P Figure 10.
Pipe equator obtained by a projection to the unit discand applying a homotopy there.Now that we have all the pieces in place, we can construct the homotopy ofembeddings, one hemisphere at a time. Denote by S := { ( θ, φ ) ∈ S : φ > } , andby D the space of smooth simple closed loops in D such that they divide the areaof D in half. To define a homotopy G : [0 , × S → E q + we require the followingmap ˜ f : E q + → D induced by f , namely L f ( L ). Define, G t ( θ, φ ) = ˜ f − ◦ F ,θ ◦ ˜ f ◦ i ( θ, φ ) , φ ∈ [0 , ǫ ] ,F [ t · φ − ǫǫ ,θ ] ◦ ˜ f ◦ i ( θ, φ ) , φ ∈ [ ǫ, ǫ ] ,F t,θ ◦ ˜ f ◦ i ( θ, φ ) , φ ∈ [2 ǫ, π − ǫ ] ,F [ t · ( π/ − ǫ ) − φǫ ,θ ] ◦ ˜ f ◦ i ( θ, φ ) , φ ∈ [ π − ǫ, π − ǫ ] ,F ,θ ◦ ˜ f ◦ i ( θ, φ ) , φ ∈ [ π − ǫ, π ) , where, by F • ,θ ◦ ˜ f ◦ i we mean to apply F • ,θ to the simple closed curve given by˜ f ◦ i . As discussed earlier, F • ,θ preserves the area enclosed by ˜ f ◦ i . Thus, G t iswell defined. Evidently, G ≡ i , and G embeds S in E q + , up to ǫ and δ . Notethat ǫ is chosen such that G transitions continuously from i to the pipe equatorembedding. Such a transition is required since the latter embedding cannot bedefined for sufficiently small φ , due to the area required for the pipe, as explainedin Subsection 1.3.For φ ∈ ( − π/ , φ ց
0, both D and D shrink in area. For φ = 0 both caps collapse to a point. Then as φ descends below 0, D becomes thecap which is always covered by hemispheres as θ varies, whereas D becomes thecap which is never obtained by the hemispheres. Now, pick x φ to be the center of D , puncture the sphere at x φ to obtain h : S \ { x φ } → D , and ˜ h : E q + → D ,analogously to f and ˜ f above.Define H : [0 , × S − → E q + , by H t ( θ, φ ) = ˜ h − ◦ F ,θ ◦ ˜ h ◦ i ( θ, φ ) , φ ∈ [ − ǫ, ,F [ − t · ǫ + φǫ ,θ ] ◦ ˜ h ◦ i ( θ, φ ) , φ ∈ [ − ǫ, − ǫ ] ,F t,θ ◦ ˜ h ◦ i ( θ, φ ) , φ ∈ [ − π + ǫ, − ǫ ] ,F [ t · φ +( π/ − ǫ ) ǫ ,θ ] ◦ ˜ h ◦ i ( θ, φ ) , φ ∈ [ − π + 2 ǫ, − π + ǫ ] ,F ,θ ◦ ˜ h ◦ i ( θ, φ ) , φ ∈ ( − π , − π + 2 ǫ ] , where S − := { ( θ, φ ) ∈ S : φ < } . Now we may define the following amalgamationof G and H as A : [0 , × S → E q + , by A t ( θ, φ ) = ( H t ( θ, φ ) , φ ∈ ( − π , ,G t ( θ, φ ) , φ ∈ (0 , π ) . Observe that for all t ∈ [0 ,
1] and all φ ∈ ( − ǫ, ǫ ) it holds that A t ≡ i . Since i iscontinuous, so is A for these values. For all other φ , A coincides with either G or H and is therefore continuous.This completes the construction of the homotopy between the natural embedding i and a pipe equator embedding i p associated with δ and ǫ .3. The Hofer diameter of a pipe equator embedding
In this section we provide an upper bound on the Hofer distance between twopipe equators, represented by points away from the boundary of the parametersspace. This is done by considering a slightly more convenient planar setting, wheretwo types of flows are applied. The upper bound is then given by calculating theenergy cost of applying these flows in two possible ways.
N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 13
First, let us describe the dynamical approach to the formation of pipe equators,which will then be used as part of the planar picture. Consider S , and constructa thin strip of area 0 < δ ≪ / L to itself. As before, we refer to this strip as a pipe, and denote thearea to one of its sides in the upper hemisphere by a . Note that while the choiceof a side whose area we denote is arbitrary, it does not affect the following, as a isuniquely determined by the area on the other side of the pipe.Given such a pipe, we flow the lower hemisphere through it via a Hamiltoniandiffeomorphism, reasoning similarly to constructions given in Sections 2.1 and 2.2in [Kha15]. Let H be an autonomous Hamiltonian such that on one of the pipe’ssides H = 0, on its other side H = 1, and H is linearly interpolated in between;see Figure 11, where the regions in blue are where H is constant. Apply a C ∞ L Figure 11.
Hamiltonian diffeomorphism creating a pipe equator,and a few flow lines.smoothing to H near the singular points. Denote by φ tH the time- t map of the flowof H . As explained in [Kha15], after time t + ǫ , a region of the lower hemisphere ofarea t will be flowed along the pipe, where ǫ depends on the choice of smoothing of H and can be made arbitrarily small. Given that the lower hemisphere has beenflowed for time t + ǫ > δ , the area of the pipe is pushed in the lower hemisphere toform a strip of area δ , which in our terminology is also called a pipe. Observe thatthe time parameter of the flow uniquely determines the area of the lower hemispherenot yet flowed through the pipe by b = 1 / − t − δ . This area is the second parameterwe mentioned of the resulting pipe equator. Hence, in the dynamical approach wedefine the pipe equator associated with a, b , as the image of φ tH ( L ), where t = t ( b ),and H depends on a .Note that in Section 2 we used angle parameters to describe pipe equators,whereas here we use area parameters. By the construction of the pipe equatorembedding, we may translate one to the other. Observe that a = Area( S ) in thenotation of Section 2, for which we provided a formula for φ ∈ (0 , π/ − ǫ ). Hence,combining with the corresponding formula for negative φ , we obtain, a ( φ ) = 2 δ − π − ǫ φ + 14 − δ , As mentioned, b is the area not yet flowed through the pipe, and by constructionof the pipe equator embedding is chosen to depend on θ linearly; namely, b ( θ ) = 2 δ − π θ + 12 − δ. Let us now describe the planar outlook. In the notation of the previous section,since we fixed δ and ǫ , the parameter φ is bounded away from 0 and there is somecap, D , unswept by the flow of L . Observe that U := S \ D is open, and maybe considered as a neighborhood containing all pipe equators associated with δ .By restricting the discussion to U and changing coordinates, we may consider pipeequators as curves in the plane, as in Figure 12. For ease of notation we denotethis restriction to U and change of coordinates as a map q : U → R . i p ( θ , φ ) D P e U − a − δ Figure 12.
Pipe equator projected to the plane. Note the colorsmatch Figure 3.Let us clarify the notations of Figure 12. Let γ be a pipe equator as the imageof L under a Hamiltonian flow φ , such that its image under q is the red curve inthe figure. Denote by H the lower hemisphere of S , and P the pipe in the upperhemisphere. Denote by H r the region of H that has not already been flowedthrough P , and by H f the region that has been flowed. Thus, in terms of Figure12, L denotes the image q ( H r ), similarly R denotes q ( H f ). Likewise, denote theupper hemisphere of S by H , and analogously H f , and H r , as the regions thathave (and have not) been flowed through the pipe in the lower hemisphere. Applythe change of coordinates q , so C denotes q ( H r ), and U denotes q ( H f ) in termsof Figure 12. Finally, let P e denote the image under q of the pipe in the upperhemisphere, and P i denote the image under q of the pipe in the lower hemisphere.As the pipe equator γ is determined by the areas a and b , so are the areas in theplane. Namely, L has area b , C has area a , the pipes both have area δ , and R and U have complementing areas to L , C , and the pipes.Therefore, given two pipe equators γ = i p ( a , b ), γ = i p ( a , b ) as above,consider their Hofer distance(4) d H ( i p ( a , b ) , i p ( a , b )) , as a function of 0 < a j , b j < /
2, for j = 1 ,
2. In the following we bound thisdistance from above by estimating the symplectic energy it would take to deformone to the other under q . At the beginning of this section, we used arguments from[Kha15] to estimate the energy it takes to construct a pipe equator on the sphere. N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 15
In fact, the same arguments can be used to deform equators to one another byflowing the regions they enclose. We consider two specific such deformations in theplane. First, apply a flow whose Hamiltonian is as depicted in Figure 13, in orderto flow the region R towards L in terms of Figure 12. This flow may also be appliedin negative time to flow L towards R . D H = 1 H = 0 Figure 13.
Flowing equators in the disc.The second deformation we consider, consists in flowing the regions C and U , interms of Figure 12. By applying a flow associated with a Hamiltonian described inFigure 14, the region outside the disk D is flowed through the pipe P i , such thatthe external pipe P e is deformed to enclose a smaller area between D and itself.Intuitively speaking, this flow shrinks the area between the external pipe and thedisc. D H = 0 H = 1 Figure 14.
Shrinking the areas enclosed by the pipe and the disc.Given two equators as in Figure 15 we use the above deformations in order todescribe two approaches to deform one equator into the other. Roughly speaking,the first approach is to flow each of the two equators such that they both coincide with ∂ D . The second approach is to flow the overlapping regions between theequators to make them coincide. D γ γ Figure 15.
Two pipe equators in general position projected to the plane.In the first approach, consider each pipe equator separately and observe thatthere are four different manners in which an equator can be flowed through thepipes such that it coincides with ∂ D . In the notation of Figure 12: the first (andsecond) possibility is to flow L ( R ) through P e and then flow P e itself; the third(and fourth) possibility is to flow C ( U ) through P i and then flow P i itself. Sinceboth pipes have area δ , and in all four cases the pipes are also flowed, we calculatethe required energy only up to δ and do not include the pipes in the argumentsbelow. Flowing L is done with energy cost of b , as explained above; likewise, R costs 1 / − b to flow, C costs a , and U costs 1 / − a . Since we are interested inminimizing the cost, this yields the following upper bound on deforming a givenpipe equator i p ( a, b ) to ∂ D , min (cid:26) a, − a, b, − b (cid:27) . Therefore, it follows that the first approach yields the following cost function, f ( a , b , a , b ) = min (cid:26) a , − a , b , − b (cid:27) + min (cid:26) a , − a , b , − b (cid:27) . In the second approach, instead of deforming both equators to coincide with ∂ D , we deform them to coincide with each other. This consists in flowing L tocoincide with L , or the other way around; and in flowing C to coincide with C ,or conversely C to coincide with C . Of course, flowing L to L is the same as R to R , and the same holds for the relation between the C ’s and U ’s. Hence, thecost of the former is | b − b | , while the cost of the latter is | a − a | . Overall, thesecond approach yields the following cost function, f ( a , b , a , b ) = | a − a | + | b − b | . So, altogether given two equators L = i p ( a , b ), L ( a , b ), the following holdsfor their Hofer distance d H ( L , L ) ≤ min { f ( a , b , a , b ) , f ( a , b , a , b ) } . N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 17
Therefore, put F := min { f , f } . It thus follows that diam( i p ) ≤ max F, whence it remains to maximize F .A maximum of F in [0 , / is an upper bound for (4), up to δ >
0. We claimthat F ≤ /
3, and that this value is attained at x = (1 / , / , / , / . Clearly, F ( x ) = 1 /
3, whence x is a maximal point for F . Note that x is notunique, there are at least 3 other permutations of the coordinates of x which yielda maximum for F .To see that F ≤ /
3, first consider x = ( a , b , a , b ) ∈ [1 / , / . It holds that | a − a | ≤ /
6, and likewise | b − b | ≤ /
6, so f ( x ) ≤ /
3. Therefore, F ( x ) = min { f ( x ) , f ( x ) } ≤ f ( x ) ≤ . Second, for x = ( a , b , a , b ) outside of [1 / , / , consider f ( x ). It holds that m j ( x ) := min (cid:26) a j , − a j , b j , − b j (cid:27) ≤ , for j = 1 ,
2. Indeed, for each j = 1 ,
2, pick any of the terms in the minimumexpression above, and denote it by c . By our assumption, either c ≤ / / − c ≤ /
6. Therefore, m j ( x ) ≤ /
6, and so F ( x ) = min { f ( x ) , f ( x ) } ≤ f ( x ) = m ( x ) + m ( x ) ≤ / , as claimed.Finally, remark that all “non-pipe” equators described by the pipe equator em-bedding (i.e., i p ( θ, φ ) for φ ∈ [ − π/ , − π/ ǫ ] ∪ [ π/ − ǫ, π/ ǫ -close to L . Hence, up to ǫ , the estimates above hold for them as well. Thisfinishes the estimate of the upper bound in Theorem 1.4. Non-triviality of [ i ]Observe that Ham( S ) acts on E q + since Hamiltonian diffeomorphisms preserveareas. Denote S = Stab Ham( S ) ( L ) the stabilizer of the standard equator L withpositive orientation. We have the following fibration(5) S Ham( S ) E q + This is true since Ham( S ) is a Lie group, and so Ham( S ) → Ham( S ) /S is afibration with fiber S ; see, e.g. [Ste99] Sections 7.3-7.5. Here we used the factthat E q + is homeomorphic to the orbit space Ham( S ) /S ; this is true because theaction is transitive, indeed every equator is obtained from L by applying someHamiltonian diffeomorphism. Given the fibration above, we can consider the long exact sequence it induces onhomotopy groups (see, e.g., Theorem 4.41 in [Hat02]),(6) π (Ham( S ) , ) → π ( E q + , L ) ∂ −→ π ( S, ) → π (Ham( S ) , ) → π ( E q + , L ) . It holds that Ham( S ) is homotopy equivalent to SO (3). This is due to the factthat Ham( S ) = Symp ( S ) (see, e.g., Section 1.4 in [Pol01]), and since Diff + ( S ) ⊃ Symp ( S ) has SO (3) as a strong deformation retract, by Theorem A in [Sma59].Since SO (3) has the homotopy type of R P , it follows that, π (Ham( S ) , ) ∼ = π (Ham( S )) ∼ = π ( R P ) = 0 , and similarly, π (Ham( S ) , ) ∼ = π ( R P ) = Z / Z . As for the π ( S, ) in (6), consider the following two maps. First, i : S → S given by t R (2 πt, ˆ z ), where R is described in the notation of Subsection 1.2, ˆ z isa unit vector in the direction of the z -axis, and S = R / Z . The second map is theevaluation ev : S → S given by φ l − ( φ ( l (0))), where l : S → L is some fixedparametrization of L . We have the following commutative diagram: S S S i ev This diagram descends to a diagram of fundamental groups. π ( S , π ( S, ) π ( S , i ∗ ev ∗ Hence it follows that i ∗ is injective, and so π ( S, ) contains a Z subgroup.Going back to (6), we have the following excerpt of an exact sequence:(7) π ( E q + , L ) π ( S, ) Z / Z q p Since Z < π ( S, ), the map p in (7) cannot be injective. By exactness at π ( S, )it follows that Image( q ) = 0. This shows that π ( E q + , L ) = 0.Finally, observe that [ i ] = 0 ∈ π ( E q + , L ). To see this, note that its imageunder the connecting morphism ∂ : π ( E q + , L ) → π ( S, ) in (6) is also non-trivial.Indeed, recall that by (5) we have a fibration Ham( S ) → E q + . One may consider i as a map ( D , ∂ D ) → ( E q + , L ), by collapsing the S boundary of D to apoint. By the homotopy lifting property of fibrations, it follows that there existsa lift ˜ i : ( D , ∂ D ) → (Ham( S ) , S ). Since the boundary ∂ D of the disk mapsto L , it follows that the image of ˜ i | ∂ D is contained in the fiber S . Therefore,one may consider ˜ i | ∂ D as a map ( S , → ( S, ). By definition of the connectingmorphism, it holds that(8) ∂ [ i ] = [˜ i | ∂ D ] . We claim now that [˜ i | ∂ D ] is not a unit, and by (8) it follows that [ i ] is not aunit either. Observe that a lift of i to Ham( S ) may be chosen to have its imagecontained in SO(3). Since SO(3) is diffeomorphic to S S , the unit circle bundle in T S , it follows that a choice of a lift for i : D → SO(3) is equivalent to a choice
N THE HOFER GIRTH OF THE SPHERE OF GREAT CIRCLES 19 of a section v ∈ Γ( S S ), such that it has a singular point at N , the north pole of S . After multiplying v by a function λ : S → R which vanishes exactly at N , wemay apply the renowned Poincar´e-Hopf Theorem, to obtainindex N ( λ · v ) = χ ( S ) = 2 , where χ ( S ) is the Euler characteristic of the sphere, which is well known to be 2.Now, recall that ev ∗ denotes the map π ( S, ) → π ( S ,
0) induced by evaluation.By definition of the index, and considering π ( S , ∼ = Z , we find thatev ∗ [˜ i | ∂ D ] = 2 . Notably, it follows that [˜ i | ∂ D ] = 0, as claimed. References [Che00] Yu. V. Chekanov,
Invariant Finsler metrics on the space of Lagrangian embeddings ,Math. Z. (2000), no. 3, 605–619. MR1774099[Gro96] Mikhael Gromov,
Systoles and intersystolic inequalities , Actes de la Table Ronde deG´eom´etrie Diff´erentielle (Luminy, 1992), 1996, pp. 291–362. MR1427763[Hat02] Allen Hatcher,
Algebraic topology , Cambridge University Press, Cambridge, 2002.MR1867354[Kha09] Michael Khanevsky,
Hofer’s metric on the space of diameters , J. Topol. Anal. (2009),no. 4, 407–416. MR2597651[Kha15] , Hofer’s length spectrum of symplectic surfaces , J. Mod. Dyn. (2015), 219–235.MR3395269[LS14] Fran¸cois Lalonde and Yasha Savelyev, On the injectivity radius in Hofer’s geometry ,Electron. Res. Announc. Math. Sci. (2014), 177–185. MR3356596[MS17] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology , Third, OxfordGraduate Texts in Mathematics, Oxford University Press, Oxford, 2017. MR3674984[Pol01] Leonid Polterovich,
The geometry of the group of symplectic diffeomorphisms , Lecturesin Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel, 2001. MR1826128[Sav15] Yasha Savelyev,
Global Fukaya category and the space of A ∞ categories II (2015), avail-able at arXiv:1408.3250[math.SG] .[Sma59] Stephen Smale, Diffeomorphisms of the -sphere , Proc. Amer. Math. Soc. (1959),621–626. MR112149[Ste99] Norman Steenrod, The topology of fibre bundles , Princeton Landmarks in Mathematics,Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition, PrincetonPaperbacks. MR1688579
Itamar Rosenfeld Rauch, Mathematics Department, Technion - Israel Institue ofTechnology, Haifa, 32000, Israel
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