Lens rigidity with trapped geodesics in two dimensions
aa r X i v : . [ m a t h . DG ] A ug LENS RIGIDITY WITH TRAPPED GEODESICS IN TWODIMENSIONS
CHRISTOPHER B. CROKE + AND PILAR HERREROS † Abstract.
We consider the scattering and lens rigidity of compact surfaceswith boundary that have a trapped geodesic. In particular we show thatthe flat cylinder and the flat M¨obius strip are determined by their lens data.We also see by example that the flat M¨obius strip is not determined by it’sscattering data. We then consider the case of negatively curved cylinders withconvex boundary and show that they are lens rigid. Introduction
In this paper we consider the lens and scattering rigidity of a number of compactsurfaces with boundary that have a trapped geodesic. A trapped geodesic ray is ageodesic γ ( t ) which is defined for all t ≥
0, while a trapped geodesic is one definedfor all t . We will call a unit vector trapped if it is tangent to a geodesic ray whilewe call it totally trapped if it is tangent to a trapped geodesic.We will consider compact two dimensional manifolds ( M, ∂M, g ) with boundary ∂M and metric g . Let U + ∂M represent the space of inwardly pointing unit vectorsat the boundary. That is, v ∈ U + ∂M means that v is a unit vector based at aboundary point and h v, η + i ≥
0, where η + is the unit vector of M normal to ∂M and pointing inward. U − ∂M will represent the outward vectors. These spaces aretwo dimensional while U + ∂M ∩ U − ∂M = U ( ∂M ) the unit tangent bundle of ∂M is one dimensional.For v ∈ U + ∂M let γ v ( t ) be the geodesic with γ ′ (0) = v . We let T T ( v ) ∈ [0 , ∞ ](the travel time) be the first time t > γ v ( t ) hits the boundary again. If γ v ( t )never hits the boundary again then T T ( v ) = ∞ , while if either γ v ( t ) does not existfor any t > t > γ ( t ) ∈ ∂M ,then we let T T ( v ) = 0. Note that T T ( v ) = 0 implies that v ∈ U ( ∂M ) while for v ∈ U ( ∂M ), T T ( v ) may or may not be 0.The scattering map S : U + ∂M → U − ∂M takes a vector v ∈ U + ∂M to thevector γ ′ ( T T ( v )) ∈ U − ∂M . It will not be defined when T T ( v ) = ∞ and will be v itself when T T ( v ) = 0. If another surface ( M , ∂M , g ) has isometric boundary to( M, ∂M, g ) in the sense that ( ∂M, g ) ( g restricted to ∂M ) is isometric to ( ∂M , g )(i.e. they have the same number of components - circles - with the same lengths),then we can identify U + ∂M with U + ∂M and U − ∂M with U − ∂M . We say that( M, ∂M, g ) and ( M , ∂M , g ) have the same scattering data if they have isometricboundaries and under the identifications given by the isometry they have the same Key words and phrases.
Scattering rigidity, Lens rigidity, trapped geodesics.+Supported in part by NSF grant DMS 10-03679 and an Eisenbud Professorship at M.S.R.I.. † Supported in part by an M.S.R.I. Postdoctoral Fellowship.
Figure 1.
Not isometric but same scattering and lens data.scattering map. If in addition the travel times
T T ( v ) coincide then they are saidto have the same lens data.A compact manifold ( M, ∂M, g ) is said to be scattering (resp. lens) rigid if forany other manifold ( M , ∂M , g ) with the same scattering (resp. lens) data thereis an isometry from M to M that agrees with the given isometry of the boundaries.In this paper we prove three such rigidity results. Theorem 1.1.
The flat cylinder [ − , × S is lens rigid. Theorem 1.2.
The flat M¨obius strip is lens rigid.
Theorem 1.3.
A cylinder with negative curvature and convex boundary is lensrigid.
The higher dimensional version of theorem 1.1 was proved recently [Cr11] by thefirst author. In that paper it was shown that for n ≥ D n × S is scattering rigidwhere D n represents the unit disc in R n . This was the first example of such a rigiditytheorem that had trapped geodesic rays (however [St-Uh09] has a local rigidityresult that includes trapped geodesic rays). The two dimensional case has a numberof differences from the higher dimensional case. Although it is possible to approachTheorem 1.1 with methods as in [Cr11] there are a number of complications. Inparticular, the boundary is neither connected nor does the second fundamentalform have a positive eigenvalue. Here we use a different approach entirely, which isvery two dimensional and also allows us to prove the other results. We should notethat in the two dimensional case we do not prove scattering rigidity, but only lensrigidity. We see by example (see below) that the flat M¨obius band is not scatteringrigid (at least if one allows C metrics) while the other two cases are still open.The fact that not all manifolds are scattering rigid was pointed out in [Cr91].For > ǫ > h ( t ) be a small smooth bump function which is 0 outside ( − ǫ, ǫ )and positive in ( − ǫ, ǫ ). For s ∈ ( − ǫ, − ǫ ) consider surfaces of revolution g s with smooth generating functions F s ( t ) = 1 + h ( s + t ) for t ∈ [ − , s but otherwise look the same (see figure 1). The Clairaut relations show that,independent of s , geodesics entering one side with a given initial condition exitout the other side after the same distance at the same point with the same angle. ENS RIGIDITY WITH TRAPPED GEODESICS IN TWO DIMENSIONS 3
PSfrag replacements M CH α αα α Figure 2.
Same scattering but not lens data.Hence all metrics have the same scattering data (and in fact lens data) but are notisometric. A much larger class of examples was given in section 6 of [Cr-Kl94].We now present an example that shows that the flat M¨obius band is not scat-tering rigid. Let C be the cylinder [0 , l ] × S and let H be a hemisphere attachedto C by identifying the equator with the the curve l × S . We get M = C ∪ H/ ∼ where ∼ is the identification above. Note that M is topologically a disc.We need to understand some of the geodesics on M . Observe that any geodesicin the cylinder that reaches l × S forming an angle α with it goes into H , where itis a great circle that leaves H again at its antipodal point forming the same angle α .From the point of view of the cylinder, any geodesic that leaves it through a point( l, θ ) comes back at the point ( l, θ + π ) with the same angle. Thus, the scatteringdata of M is the same as that of M ; the cylinder with one boundary identifiedto itself via the antipodal map. I.e. M is a flat M¨obius band. Therefore, thescattering data of M and M are the same, but the travel times are different. Infact they differ by exactly π .All known examples of nonisometric spaces with the same lens or scattering datahave in common that there are trapped geodesics.The scattering and lens rigidity problems are closely related to other inverseproblems. In particular, the boundary rigidity problem is equivalent to the lensrigidity question in the Simple and SGM cases. See [Cr91] and [Cr04] for definitionsand relations to some other problems. There is a vast literature on these problems(see for example [Be83, Bu-Iv06, Cr91, Cr90, Gr83, Mi81, Mu77, Ot90-2, Pe-Sh88,Pe-Uh05]). In particular, it was shown in [Pe-Uh05] that Simple two dimensionalcompact manifolds are boundary rigid (hence lens and scattering rigid). The Simplecondition however precludes trapped geodesic rays.The main issue in the proofs of all the Theorems in this paper is to show thatthe space of trapped geodesics has measure 0. We will get at this by countingintersections of geodesics and applying a version of Crofton’s formula. We do thisin Section 2.We prove Theorems 1.1 and Theorem 1.2 in section 3 using rigidity argumentsdeveloped in [Cr91] and [Cr-Kl98]. Theorem 1.3 is proved in section 4 using arigidity method developed by Otal in [Ot90-1, Ot90-2]).2. Counting Intersections
In this section we discuss how to use a version of Crofton’s formula to show thattrapped geodesics have measure 0.
C. CROKE AND P. HERREROS
We begin with the general case of two 2-dimensional manifolds M and M withthe same boundary and the same scattering data.The space of geodesics that start at the boundary can be parametrised by theirinitial vector in U + ∂M . For s ∈ ∂M and θ ∈ [ − π , π ] let γ ( s,θ ) be the geodesicstarting at s that makes an angle θ with the inward direction. The Liouville measureon the space of geodesics leaving the boundary can be represented as | cos( θ ) | dθds ,where ds represents the arclength along the boundary. In fact, Santal´o’s formula(see chapter 19 of [Sa76]) tells us that this is true for any curve τ in M . Namely, ifwe parametrise the geodesics passing through τ by arclength dt along τ and angle φ made with a chosen normal, then the Liouville measure will be | cos( φ ) | dφdt . Ofcourse γ ( s,θ ) might intersect the curve τ many times (or not at all). Let i ( τ, s, θ )be the geometric number of times that γ ( s,θ ) intersects τ . Also let G ( τ ( t )) be thesubset of the unit vectors at τ ( t ) that are tangent to geodesics that started at aboundary point. The above gives us the following version of Crofton’s formula(which works in both M and M ): Z ∂M Z π − π i ( τ, s, θ ) | cos( θ ) | dθds = Z τ Z G ( τ ( t )) | cos( φ ) | dφdt. We will let Γ (resp Γ ) be the space of non-trapped geodesics that are not tangentto the boundary at either endpoint. Γ can be parameterized as an open subset ofthe unit vectors U + ∂M on the boundary pointing inward. Γ can be identifiedwith Γ by this parametrization. We will consider the corresponding intersectionfunctions i ( γ, τ ) and i ( γ , τ ) which map Γ × Γ − Diag to the nonnegative integersvia the geometric intersection number (i.e. the number of intersection points) ofthe geodesics γ and τ (respectively γ and τ ), where γ and τ are distinct nontrapped geodesics (running from boundary point to boundary point) of M and γ and τ are the corresponding geodesics in M . We will show that these functionsare closely related. They need not be the same though as the counter example toscattering rigidity for the M¨obius strip has i = i + 1. Lemma 2.1.
Let γ , τ and τ be distinct elements of Γ such that τ and τ arein the same component of Γ . Then i ( γ, τ ) − i ( γ , τ ) = i ( γ, τ ) − i ( γ , τ ) . Proof.
Since Γ is an open subset of a 2-dimensional manifold we can (by standardtransversality arguments) choose a smooth path τ t from τ to τ such that τ t = γ and τ t = − γ for any t ∈ [0 , τ t intersects transversely the subspace End ( γ )of Γ consisting of geodesics with an endpoint in common with γ . In particular, if anendpoint of τ t (say τ t (0)) coincides with an endpoint of γ , then W = ddt | t τ t (0)is not the zero vector. Since geodesics always intersect transversely (except atboundary points) f ( t ) = i ( γ, τ t ) (resp f ( t ) = i ( γ, τ t )) only changes for those t ’swhen τ t ∈ End ( γ ). As we pass trough t f ( t ) and f ( t ) change by exactly 1 (eitherplus or minus). However the sign of the change is determined by W (more precisely,the direction on the boundary determined by W ) and the inward tangents to γ and τ at the common boundary point. That is, if the inward tangent to γ lies between W and the inward tangent to τ t , then both f and f increase by one and they willdecrease by one otherwise. In either case we see that f ( t ) − f ( t ) is constant. (cid:3) We will apply this lemma to our various cases. In the case of the flat M¨obiusstrip Γ is connected and hence i = i + n for some integer n . However, there are ENS RIGIDITY WITH TRAPPED GEODESICS IN TWO DIMENSIONS 5 geodesics γ and τ in M that don’t intersect at all so 0 ≤ i ( γ , τ ) = 0 + n . Hence n is a nonnegative integer. In the case of the flat torus Γ has two components,but since one component is gotten from the other by reversing orientations of thegeodesics, and since intersection numbers are independent of orientation, we againconclude i = i + n where as before n is a nonnegative integer.Consider the case of a negatively curved cylinder with convex boundary withboundary components ∂ and ∂ . It is straightforward to see that (up to reversingorientations) there are three components: Those geodesic going from ∂ to ∂ ; thosegoing from ∂ to ∂ ; and those going from ∂ to ∂ . However, for any pair of suchcomponents (including when both are the same component) we can find a geodesicfrom each component that do not intersect each other. The previous argument thentells us that i ≥ i .Our next goal is to study the measure of the set of trapped geodesics. To thatend, for a surface M with boundary, we let T G + ( x ) ⊂ U x be the set of unit vectors v at x ∈ M such that the geodesic ray in the v direction never hits the boundary.Further we define T G − = { v |− v ∈ T G + } , T G ( x ) = T G + ( x ) ∪ T G − ( x ) (the trappeddirections), and T T G ( x ) = T G + ( x ) ∩ T G − ( x ) (the totally trapped directions). Wesay the space of trapped geodesics has measure 0 if the measure of T G ( x ) is 0 forall x . Lemma 2.2.
Let M and M be surfaces with the same scattering data and γ ∈ Γ .Assume that the space of trapped geodesics in M has measure . If for every τ ∈ Γ we have i ( γ, τ ) ≤ i ( γ , τ ) then L ( γ ) ≤ L ( γ ) . Further if L ( γ ) = L ( γ ) then T G ( γ ( t )) has measure for almost all t .Proof. First note that4 L ( γ ) = Z L ( γ )0 Z π | cos ( θ ) | dθdt ≥ Z L ( γ )0 Z G ( γ ( t )) | cos ( θ ) | dθdt. While Crofton’s formula says Z L ( γ )0 Z G ( γ ( t )) | cos ( θ ) | dθdt = Z Γ i ( γ , τ ) dτ ≥ Z Γ i ( γ, τ ) dτ = 4 L ( γ ) . In the above we used that the measures dτ and dτ on Γ are the same. In orderfor equality to hold not only must i ( γ, · ) and i ( γ, · ) coincide but T G ( γ ( t )) musthave measure 0 for almost all t . (cid:3) The flat case
In this section we will prove Theorems 1.1 and 1.2. We will start by consideringthe cylinder case. Let M = [0 , × S be a flat cylinder and suppose ( M , ∂M , g )is a surface with the same lens data as M .We see that the geodesics that start perpendicular to the boundary (and henceend perpendicular to the boundary) all have length 1 and achieve the distancebetween the boundary components. In particular they are minimizing geodesics,no two intersect and the union covers M (since a shortest path from any interiorpoint of M to the boundary will hit the boundary perpendicularly). Thus thereis a natural diffeomorphism F : M = { ( t, θ ) ∈ [0 , × S } → M . Along thegeodesic γ θ of M that starts perpendicular to the boundary at (0 , θ ) the vector C. CROKE AND P. HERREROS field ddθ = j ( t, θ ) N ( t, θ ) (where N ( t ) is the unit vector field in the ddθ direction) isa Jacobi field perpendicular to γ θ . By the above Area ( M ) = Z S Z j ( t, θ ) dtdθ. The fact that M has the same lens data as M says that Jacobi fields along γ θ correspond to those along γ θ in M in the sense that, if some Jacobi field J along γ θ has the same initial conditions (value and covariant derivative) as a Jacobi field J along γ θ , then they also must have the same final conditions. This being true forall Jacobi fields along γ θ is equivalent (see [Cr91]) to Z j − ( t, θ ) dt = Z dt = 1 . But the convexity of f ( x ) = x − tells us that R dt = R j − ( t, θ ) dt ≥ { R j ( t, θ ) dt } − with equality if and only if j ( t, θ ) ≡
1. And hence we see that
Area ( M ) = Z S Z j ( t, θ ) dtdθ ≥ Z S Z dtdθ = Area ( M )with equality holding if and only if j ( t, θ ) ≡
1, i.e. M is isometric to M with theisometry being the diffeomorphism F described above. Thus we have shown Lemma 3.1.
Let M be a flat cylinder . Then if M is a surface with the same lensdata then Area ( M ) ≥ Area ( M ) with equality holding if and only if M is isometric to M . On the other hand we have shown in the previous section that the set of unitvectors in M tangent to trapped geodesic rays has measure 0. (This is of coursealso true of M .) Now Santal´o’s formula and the invariance of the Liuoville measureunder the geodesic flow tells us that the Liouville volume of the unit tangent bundleof M (resp. M ) is R U + ∂M L ( γ ( v )) dv (respectively R U + ∂M L ( γ ( v )) dv ), where themeasures dv = | cos( θ ) | dθds on U + ∂M and U + ∂M are the same. Thus the lensequivalence tells us that the unit tangent bundle of M has the same measure asthat of M and hence the areas are the same (see chapter 19 of [Sa76]). Thus weconclude the isometry of M and M , which completes the proof of Theorem 1.1.We now consider the M¨obius strip case. We want to do this by passing to theorientation double cover of M and M and then apply Theorem 1.1. The onlyreal issue in doing this is to see that M is not orientable. (Note that in thecounterexample to scattering rigidity M is orientable.) The key point to noteis that the argument in the previous section says that the geodesics leaving theboundary perpendicularly cannot intersect (or else they would be too long). Thusin M going across such a geodesic and following the boundary back to the originalpoint reverses orientation (just as in M ). Thus we can pass to the two fold coversto complete the proof of Theorem 1.2. ENS RIGIDITY WITH TRAPPED GEODESICS IN TWO DIMENSIONS 7 Negative curvature
In this section we will prove Theorem 1.3.Fix a boundary point x ∈ ∂M and its corresponding point x ∈ ∂M . Let τ : ( −∞ , ∞ ) → ∂M be the unit speed parametrization of the boundary componentwith τ (0) = x (which of course goes around the boundary infinitely often). Similarlydefine τ . We let γ t be the geodesic segment (varying continuously in t ) from x to τ ( t ). Let γ t be the corresponding geodesic segment in M .Our first goal is to show that there are no conjugate points along any geodesic in M . By the convexity of the boundary, for t near 0 both γ t and γ t are minimizing.In particular, for small t there are no conjugate points along γ t . If any such geodesic γ t has a conjugate point let t be the first t (i.e. | t | is the smallest) where thishappens. Since γ t is a smooth variation, the conjugate pair must be the endpoints.However, the lens equivalence would imply that the endpoints are also conjugatealong γ t , but this can’t happen by the negative curvature assumption. This coversall geodesics from this boundary component to itself. Of course a similar argumentworks for geodesics with both boundary points on the other component. In fact,since we also know that a minimizing geodesic between components in M willcorrespond to a minimizing geodesic in M between the components, we can usea similar continuity argument to see that there are no conjugate points along thegeodesics going from one component to the other. Now, since all geodesics leavingthe boundary are limits of geodesics that hit the boundary at both endpoints, wesee that all geodesics that start at the boundary have no conjugate points.Next we want to compare geodesics in the universal covers ˜ M and ˜ M of M and M . Thus the first step is to show that M is also a cylinder, i.e. that π ( M , x ) = Z and is generated by going once around the boundary curve, whichwe assume has length L . Using the homotopy H t = γ t ∪ − τ [0 , t ] from the trivialcurve, it follows that the geodesics γ nL are homotopic to going around the boundary n times. We also know, by the convexity of the boundary, that every homotopy classis represented by some geodesic loop at x . Thus we need only show that none ofthese loops are trivial in homotopy. However, if such a geodesic loop is contractible,then a standard minimax argument would yield a geodesic loop of index 1 whichis precluded by the no conjugate points result. This allows us to conclude thatuniversal covers ˜ M and ˜ M also have the same lens data (with the boundaries inthe universal covers identified by the covering). In particular, it now follows that allgeodesics between boundary points (and hence by taking limits all geodesics withone boundary endpoint) in ˜ M and ˜ M are minimizing. One can tell whether twogeodesics in ˜ M with disjoint endpoints on the boundary intersect simply by lookingat the endpoints. The endpoints will force the intersection number mod 2 to beeither 0 or 1. Since geodesics can intersect at most once they will intersect if andonly if this number is 1. But this means that the corresponding pair of geodesicsin ˜ M will intersect if and only if they do in ˜ M .We will need control (locally) on the covariant derivatives of the gradient of dis-tance functions from boundary points. Fix ˜ x in the interior of ˜ M with d (˜ x, ∂ ˜ M ) = d . Choose d ≥ ǫ > ǫ is less than the injectivity radius for points˜ z ∈ B (˜ x, d ). Then, by compactness, there are uniform upper and lower bounds onthe geodesic curvatures of ∂B (˜ z, ǫ ). This implies that for any ˜ y ∈ ∂ ˜ M the level setsof d (˜ y, · ) have uniformly bounded geodesic curvature at points in B (˜ x, d ). This is C. CROKE AND P. HERREROS true since for each point ˜ q on the level set and each side of the level set there is a B (˜ z, ǫ ) lying on the given side and whose boundary is tangent to the level set at ˜ q .(The two ˜ z ’s lie on the geodesic from ˜ y to ˜ q .) Thus there is a neighborhood of ˜ x and a number C such that for all ˜ y ∈ ˜ M we have |∇∇ d (˜ y, · )) | ≤ C in B (˜ x, d ). Lemma 4.1.
Let M be a cylinder of negative curvature with convex boundary. If M is a surface with the same lens data, then for every x we have T G + ( x ) consistsof at most two vectors. (Hence T G − ( x ) , and T T G ( x ) consists of at most twovectors while T G ( x ) consists of at most 4 vectors.)Proof. Fix an interior point x ∈ M . To study the set of vectors tangent to geodesicsfrom x and hitting one of the boundary components we can look to the universalcover ˜ M (whose boundary we now know has two connected components) and apoint ˜ x over x . For each point ˜ y on ∂ ˜ M there is a geodesic arc from ˜ x to ˜ y (since the minimizing path is never tangent to the convex boundary). Further thisgeodesic is unique, for if not two geodesics leaving ˜ y would intersect again - but wehave shown this doesn’t happen. Thus we get a map from ∂ ˜ M to the unit circleat ˜ x . The fact that the map is continuous follows from the fact that we have noconjugate points along geodesics that leave the boundary. Thus the unit tangentsto geodesics leaving ˜ x and hitting ∂ ˜ M come in two disjoint open intervals (onegoing to each component).Thus T G + ( x ) is the complement in the unit circle of two disjoint closed intervals.We will first see that the endpoints of these intervals vary continuously. Considerthe vectors V ˜ y (˜ x ) = −∇ d (˜ y, · ) which are tangent to the geodesic from ˜ x to ˜ y ∈ ∂ ˜ M .These vector fields (as ˜ x varies) are continuous and in a neighborhood of ˜ X haveuniformly bounded covariant derivatives by the argument in the paragraph beforethe Lemma. The endpoints of the intervals will be limits of the V ˜ y (˜ x ) as ˜ y runs offto infinity along an end of the boundary. The control we have on the derivativetells us that the vector fields V ˜ y (˜ x ) will converge to a continuous vector field.Since we know that the lengths are the same as in M , Lemma 2.2 says thatalong any geodesic γ between boundary points and for almost every t , T G + ( γ ( t ))has measure 0 and hence consists of two vectors. Thus by continuity this holds forall t . It is straightforward to see that such geodesics cover all of M . (cid:3) Note that since the totally trapped geodesics have measure 0 they are limits ofgeodesics that hit the boundary so also have no conjugate points.With these preliminaries the rest of the argument closely follows the proofsin [Ot90-1]. The assumption in that paper was that both spaces have negativecurvature (and no boundary). However, the proofs only use this fact on the targetspace, along with the facts that geodesics intersect at most once in ˜ M and ifgeodesics intersect in ˜ M then corresponding geodesics intersect in ˜ M , but we haveshown these facts above. We now outline parts of the argument here but see [Ot90-1]for more details.Consider the space ˜Γ (resp ˜Γ ) of geodesics that are not totally trapped (i.e.trapped in both directions) in ˜ M (resp. ˜ M ) with its standard (Liouville) measure.The scattering data gives a π invariant, measure preserving, homeomorphism ϕ from ˜Γ to ˜Γ.Let v ∈ U ˜ M and θ ∈ (0 , π ), denote by θv a θ rotation of v in the same fiber. If v and θv are not totally trapped, then σ v = ϕ ( γ v ) and σ θv = ϕ ( γ θv ) are geodesics ENS RIGIDITY WITH TRAPPED GEODESICS IN TWO DIMENSIONS 9 in ˜Γ that intersect at one point. Let ¯ θ ( v, θ ) be the angle at which σ θv intersects σ v .We define ¯ θ ( v,
0) = 0 and ¯ θ ( v, π ) = π . Lemma 4.2. ¯ θ is continuous, and can be continuously extended to U ˜ M × [0 , π ] .Proof. We can parameterize ˜Γ by its initial vector in U + ∂ ˜ M , then by continuityof the geodesic flow we can see that the relation between pairs of geodesics in U + ∂ ˜ M × U + ∂ ˜ M and their intersection angle is continuous, where we consider theintersection of a geodesic with itself to have angle 0 or π depending on orientation.Since the same is true in ˜ M , the function ¯ θ will be continuous when restricted tothe set where neither v nor θv is a totally trapped direction. (If a geodesic doesn’thave an initial point - i.e. is defined for all negative parameter values - and is nottrapped, it will have an endpoint on the boundary and we can define ¯ θ by reversingthe orientation.)Since ˜ M is an infinite strip with negative curvature, there is only one totallytrapped geodesic σ in ˜ M . If v ∈ U ˜ M is not totally trapped but θ v ∈ T T G , weextend ¯ θ ( v, θ ) to be the angle that σ v makes with σ . Vectors w converging to θ v either are in T G − or γ w will have basepoint in ∂ ˜ M at a distance from γ v (0)going to infinity. Therefore, σ w will have the same property and (if it converges)will converges to a geodesic in T G − , by the same argument also in T G + thereforetotally trapped. Thus the σ w converges to σ , and our extension will be continuous.If γ v is totally trapped, we can reverse the roles of v and θv . They can’t be bothtotally trapped without being the same geodesic, since totally trapped geodesicscan not intersect by Lemma 4.1. (cid:3) Note that the equivariance of the metrics on the universal cover allows us todefine ¯ θ ( v, θ ) for v ∈ U M (rather than U ˜ M ).Define the average angle asΘ( θ ) = 1 V ol ( U M ) Z UM ¯ θ ( v, θ ) dv were dv is the Liouville measure in U M . Proposition 4.3.
Θ : [0 , π ] → [0 , π ] is an increasing homeomorphism such that: (1) Θ is symmetric in π − θ . (2) Θ is super-additiveMoreover, if Θ is additive, the images under ϕ of any three geodesics that intersectat a common point, also intersect at one point. In the above (1) means Θ( π − θ ) = π − Θ( θ ) while (2) means Θ( θ + θ ) ≥ Θ( θ ) + Θ( θ ) whenever θ + θ ∈ [0 , π ]. The Proposition follows directly from theproofs in [Ot90-1, Section 2]. (Note that in that paper θ ′ is used instead of ¯ θ andΘ ′ instead of Θ.)Let F : [0 , π ] → R be a continuous convex function. By Jensen inequality, foreach value of θ F (Θ( θ )) ≤ V ol ( U M ) Z UM F (¯ θ ( v, θ )) dv. Integrating over [0 , π ] with measure sin ( θ ) dθ , and using Fubini we get Z π F (Θ( θ )) sin ( θ ) dθ ≤ V ol ( U M ) Z UM Z π F (¯ θ ( v, θ )) sin ( θ ) dθdv. Let ¯ F ( v ) = R π F (¯ θ ( v, θ )) sin ( θ ) dθ , so Z π F (Θ( θ )) sin ( θ ) dθ ≤ V ol ( U M ) Z UM ¯ F ( v ) dθdv. Lemma 4.4.
Let ( M, ∂M, g ) and ( M , ∂M , g ) be as above, and F : [0 , π ] → R any convex function. Then Z π F (Θ( θ )) sin ( θ ) dθ ≤ Z π F ( θ ) sin ( θ ) dθ. It suffices to prove that1
V ol ( U M ) Z UM ¯ F ( v ) dv = Z π F ( θ ) sin ( θ ) dθ. For this we will first average ¯ F along each nontrapped geodesic γ . Let γ = ϕ ( γ )then ϕ , which is a homeomorphism when restricted to the nontrapped geodesics, in-duces a homeomorphism from γ × (0 , π ) to γ × (0 , π ) by Φ( γ ( t ) , θ ) = ( γ (¯ t ) , ¯ θ ( γ ′ ( t ) , θ )),where γ (¯ t ) is the point of intersection. This sends the Liouville measure dλ = sin ( θ ) dθdt to d ¯ λ = sin (¯ θ ) d ¯ θd ¯ t . (Note that in the earlier sections θ represented theangle from the normal to the curve where here it represents the angle from thetangent. This is why the measure here has a sin( θ ) while before it was | cos( θ ) | ).Therefore 1 L ( γ ) Z γ ¯ F ( γ ′ ( t )) dt = 1 L ( γ ) Z γ × (0 ,π ) F (¯ θ ( γ ′ ( t ) , θ )) sin ( θ ) dθdt = 1 L ( γ ) Z γ × (0 ,π ) F (¯ θ ) sin (¯ θ ) d ¯ θd ¯ t = L ( γ ) L ( γ ) Z π F (¯ θ ) sin (¯ θ ) d ¯ θ. Since the lengths of γ and γ coincide, we have that1 L ( γ ) Z γ ¯ F ( γ ′ ( t )) dt = Z π F ( θ ) sin ( θ ) dθ along each nontrapped geodesic, and since trapped directions have measure 0, theaverage over U M is the same. Lemma 4.5. (Lemma 8 from [Ot90-1] ) Let
Θ : [0 , π ] → [0 , π ] be an increasinghomeomorphism such that (1) Θ is super-additive and symmetric in π − θ . (2) for all continuous convex function F : [0 , π ] → R Z π F (Θ( θ )) sin ( θ ) dθ ≤ Z π F ( θ ) sin ( θ ) dθ. Then Θ is the identity.Proof of Theorem 1.3. By the previous lemma Θ = Id . In particular Θ is additive,so by Lemma 4.3 the images under ϕ of any three geodesics that intersect at a pointalso intersect at one point. This determines a well defined map f : ˜ M → ˜ M thatis π invariant since ϕ is. ENS RIGIDITY WITH TRAPPED GEODESICS IN TWO DIMENSIONS 11
Let γ be a geodesic segment from the boundary to a point x ∈ M , and γ = f ( γ ) the corresponding segment in M between γ (0) and f ( x ). Since Φ( γ ( t ) , θ ) =( γ (¯ t ) , ¯ θ ( γ ′ ( t ) , θ )) sends the measure sin ( θ ) dθdt to sin (¯ θ ) d ¯ θd ¯ t , we get L ( γ ) = 12 Z γ × (0 ,π ) sin ( θ ) dθdt = 12 Z γ × (0 ,π ) sin (¯ θ ) d ¯ θd ¯ t = L ( γ ) . Therefore, the lengths of geodesics segments is preserved by f , and so it is anisometry. (cid:3) References [Be83]
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