The geodesic ray transform on two-dimensional Cartan-Hadamard manifolds
aa r X i v : . [ m a t h . DG ] D ec THE GEODESIC RAY TRANSFORM ONTWO-DIMENSIONAL CARTAN-HADAMARD MANIFOLDS
JERE LEHTONEN
Abstract.
We prove two injectivity theorems for the geodesic ray trans-form on two-dimensional, complete, simply connected Riemannian man-ifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with boundednon-positive curvature and the second with decaying non-positive cur-vature. Introduction and statement of main results
In [Hel99] Helgason presents the following result: Suppose that f is acontinuous function in R , | f ( x ) | ≤ C (1 + | x | ) − η for some η >
2, and Rf = 0where Rf is the Radon transform defined by Rf ( x, ω ) := Z R f ( x + tω ) dt for x ∈ R and ω ∈ S . Then f = 0. Since the operator R is linear thiscorresponds to the injectivity of the operator. This result was later improvedby Jensen [Jen04] requiring that f = O ( | x | − η ) , η > H : Suppose f is a continuous function on H such that | f ( x ) | ≤ C e − d g ( x,o ) , where o is a fixed point in H , and Z γ f ds = 0for every geodesic γ of H . Then f = 0.The previous results are concerned with constant curvature spaces. Thereare many related results for Radon type transforms on constant curvaturespaces and noncompact homogeneous spaces, see [Hel99],[Hel13]. Thesetypes of spaces possess many symmetries. On the other hand, there is alsoa substantial literature related to geodesic ray transforms on Riemannianmanifolds, see e.g. [Muk77], [Sha94], [PSU14]. Here the symmetry assump-tions are replaced by curvature or conjugate points conditions, but the spacesare required to be compact with boundary.In this paper we present injectivity results on two-dimensional, complete,simply connected Riemannian manifolds with non-positive Gaussian curva-ture. Such manifolds are called Cartan-Hadamard manifolds, and they are diffeomorphic to R (hence non-compact) but do not necessarily have sym-metries. In order to prove our results we extend energy estimate methodsused in [PSU13] to the non-compact case.Suppose ( M, g ) is such a manifold and we have a continuous function f : M → R . We define the geodesic ray transform If : SM → R of thefunction f as If ( x, v ) := Z ∞−∞ f ( γ x,v ( t )) dt, where the unit tangent bundle SM is defined as SM := { ( x, v ) ∈ T M : | v | g = 1 } and γ x,v is the unit speed geodesic with γ x,v (0) = x and γ ′ x,v (0) = v . Since weare working on non-compact manifolds the geodesic ray transform is not welldefined for all continuous functions. We need to impose decay requirementsfor the functions under consideration. Because of the techniques used wewill also impose decay requirements for the first derivatives of the function.We denote by C ( M ) the set of functions f ∈ C ( M ) such that for some p ∈ M one has f ( x ) → d ( p, x ) → ∞ . Suppose p ∈ M and η ∈ R . Wedefine P η ( p, M ) := { f ∈ C ( M ) : | f ( x ) | ≤ C (1 + d g ( x, p )) − η for all x ∈ M } ,P η ( p, M ) := { f ∈ C ( M ) : |∇ f | g ∈ P η +1 ( p, M ) } ∩ C ( M ) . and similarly E η ( p, M ) := { f ∈ C ( M ) : | f ( x ) | ≤ C e − ηd g ( x,p ) for all x ∈ M } ,E η ( p, M ) := { f ∈ C ( M ) : |∇ f | g ∈ E η ( p, M ) } ∩ C ( M ) . For all η > P η ( p, M ) ⊂ P η ( p, M )and E η ( p, M ) ⊂ E η ( p, M ) , which can be seen by using Lemma 2.1, equation (2.1) and the fundamentaltheorem of calculus. In addition E η ( p, M ) ⊂ P η ( p, M )for all η , η > Theorem 1.
Suppose ( M, g ) is a two-dimensional, complete, simply con-nected Riemannian manifold whose Gaussian curvature satisfies − K ≤ K ( x ) ≤ for some K . Then the geodesic ray transform is injective onthe set E η ( M ) ∩ C ( M ) for η > √ K . The second theorem considers the case of suitably decaying Gaussiancurvature. By imposing decay requirements for the Gaussian curvature weare able to relax the decay requirements of the functions we are considering.
HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 3
Theorem 2.
Suppose ( M, g ) is a two-dimensional, complete, simply con-nected Riemannian manifold of non-positive Gaussian curvature K such that K ∈ P ˜ η ( p, M ) for some ˜ η > and p ∈ M . Then the geodesic ray transformis injective on set P η ( p, M ) ∩ C ( M ) for η > . One question arising is of course the existence of manifolds satisfyingthe restrictions of the theorems. By the Cartan-Hadamard theorem suchmanifolds are always diffeomorphic with the plane R so the question is whatkind of Gaussian curvatures we can have on R endowed with a completeRiemamnian metric? The following theorem by Kazdan and Warner [KW74]answers this: Theorem.
Let K ∈ C ∞ ( R ) . A necessary and sufficient condition for thereto exist a complete Riemannian metric on R with Gaussian curvature K isthat lim r →∞ inf | x |≥ r K ( x ) ≤ . Especially for every non-positive function K ∈ C ∞ ( R ) there exists ametric on R with Gaussian curvature K .The case where the metric g differs from the euclidean metric g only insome compact set and the Gaussian curvature is everywhere non-positive isnot interesting from the geometric point of view. By a theorem of Green andGulliver [GG85] if the metric g differs from the euclidean metric g at moston a compact set and there are no conjugate points, then the manifold isisometric to ( R , g ). Since non-positively curved manifolds can not containconjugate points this would be the case.The problem of recovering a function from its integrals over all lines inthe plane goes back to Radon [Rad17]. He proved the injectivity of theintegral transform nowadays known as the Radon transform and provided areconstruction formula.It is also worth mentioning a counterexample for injectivity of the Radontransform provided by Zalcman [Zal82] He showed that on R there ex-ists a non-zero continuous function which is O ( | x | − ) along every line andintegrates to zero over any line. See also [AG93],[Arm94].This work is organized as follows. In the second section we describe thegeometrical setting of this work and present some results mostly concerningbehaviour of geodesics. The third section is about the geodesic ray trans-form. In the fourth section we derive estimates for the growth of Jacobifields in our setting and use those to prove useful decay estimates. The fifthsection contains the proofs of our main theorems. Notational convention.
Throughout this work we denote by C ( a, b, . . . ) (with a possible subscript) a constant depending on a, b, . . . The value of theconstant may vary from line to line.
Acknowledgement.
This work is part of the PhD research of the author.The author is partly supported by the Academy of Finland. The author
J. LEHTONEN wishes to thank professor M. Salo for many helpful ideas and discussionsregarding this work. The author is also thankful for J. Ilmavirta for manyinsightful comments.2.
The setting of this work and preliminaries
Throughout this paper we assume (
M, g ) to be a two-dimensional, com-plete, simply connected manifold with non-positive Gaussian curvature K .By the Cartan-Hadamard theorem the exponential map exp x : T x M → M isa diffeomorphism for every point x ∈ M . Thereby we have global normal co-ordinates centered at any point and we could equivalently work with ( R , ˜ g )where ˜ g is pullback of the metric g by exponential map, but we choose topresent this work in the general setting of ( M, g ).We make the standing assumption of unit-speed parametrization for ge-odesics. If x ∈ M and v ∈ T x M is such that | v | g = 1 we denote by γ x,v : R → M the geodesic with γ x,v (0) = x and γ ′ x,v (0) = v .The fact that for every point the exponential map is a diffeomorphismimplies that every pair of distinct points can be joined by an unique geodesic.Furthermore, by using the triangle inequality, we have(2.1) d g ( γ x,v ( t ) , p ) ≥ d g ( γ x,v ( t ) , x ) − d g ( x, p ) = | t | − d g ( x, p )for every p ∈ M and ( x, v ) ∈ SM .Because of the everywhere non-positive Gaussian curvature, the function t d g ( γ ( t ) , p ) is convex on R and the function t d g ( γ ( t ) , p ) is strictlyconvex on R for every geodesic γ and point p ∈ M (see e.g. [Pet98]).We say that the geodesic γ x,v is escaping with respect to point p if function t d g ( γ x,v ( t ) , p ) is strictly increasing on the interval [0 , ∞ ). The set of suchgeodesics is denoted by E p ( M ). Lemma 2.1.
Let p ∈ M and ( x, v ) ∈ SM . At least one of geodesics γ x,v and γ x, − v is in set E p ( M ) .Proof. The function t d g ( γ ( t ) , p ) is strictly convex on R so it has astrict global minimum. Therefore the function t d g ( γ ( t ) , p ) also has astrict global minimum, which implies that at least one of functions t d g ( γ x,v ( t ) , p ) and t d g ( γ x, − v ( t ) , p ) is strictly increasing on the interval[0 , ∞ ). (cid:3) If the geodesic γ x,v belongs to E p ( M ) equation (2.1) implies the estimate(2.2) d g ( γ x,v ( t ) , p ) ≥ ( d g ( x, p ) , if 0 ≤ t ≤ d g ( x, p ) ,t − d g ( x, p ) , if 2 d g ( x, p ) < t. The manifold M is two-dimensional and oriented and so is also the tangentspace T x M for every x ∈ M . Thus given v ∈ T x M we can define e it v ∈ T x M, t ∈ R , to be the unit vector obtained by rotating the vector v by anangle t . We will use the shorthand notation v ⊥ := e − iπ/ v . HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 5
The unit tangent bundle SM is a 3-dimensional manifold and there is anatural Riemannian metric on it, namely the Sasaki metric [Pat99]. Thevolume form given by this metric is denoted by d Σ .On the manifold SM we have the geodesic flow ϕ t : SM → SM definedby ϕ t ( x, v ) = ( γ x,v ( t ) , γ ′ x,v ( t )) . We denote by X the vector field associated with this flow. We define flows p t , h t : SM → SM as p t ( x, v ) := ( x, e it v ) ,h t ( x, v ) := ( γ x,v ⊥ ( t ) , Z ( t )) , where Z ( t ) is the parallel transport of the vector v along the geodesic γ x,v ⊥ ,and denote the associated vector fields by V and X ⊥ .These three vector fields form a global orthonormal frame for T ( SM ) andwe have following structural equations (see [PSU13])[ X, V ] = X ⊥ , [ V, X ⊥ ] = X, [ X, X ⊥ ] = − KV, where K is the Gaussian curvature of the manifold M .Let f : U ⊂ M → R be such that |∇ f | g = 1. Then level sets of thefunction f are submanifolds of M . The second fundamental form I on sucha level set is defined as I ( v, w ) := Hess( f )( v, w ) , where v, w ⊥ ∇ f and Hess( f ) is the covariant Hessian (see [Pet98]).Suppose that p ∈ M . Denote by B p ( r ) the open geodesic ball with radius r , and by S p ( r ) its boundary. Lemma 2.2.
For every p ∈ M and r > the geodesic ball B p ( r ) hasa strictly convex boundary, i.e. the second fundamental form of S p ( r ) ispositive definite.Proof. Suppose x ∈ S p ( r ) and v is tangent to S p ( r ) at x . Denote f ( y ) = d p ( y, p ). We haveHess( f )( x ) = 2 f ( x ) Hess f ( x ) + 2 d x f ⊗ d x f and thus Hess( f )( x )( v, v ) = 2 f ( x ) Hess f ( x )( v, v )since d x f ( v ) = h∇ f ( x ) , v i g = 0.Since the function t → d ( γ x,v ( t ) , p ) is strictly convex we getHess( f )( x )( v, v ) = d dt (( f ◦ γ x,v )( t )) (cid:12)(cid:12)(cid:12) t =0 > . Therefore Hess( f ) is positive definite in tangential directions and so is alsoHess f (cid:3) J. LEHTONEN
Equivalently, the boundary of B p ( r ) is strictly convex if and only if everygeodesics starting from a boundary point in a direction tangent to boundarystays outside B p ( r ) for small positive and negative times and has a secondorder contact at time t = 0. From this we see that if x ∈ M and v is tangentto S p ( d g ( x, p )) then function t d g ( γ x,v ( t ) , p ) has a global minimum at t = 0. Lemma 2.3.
Suppose p ∈ M and ( x, v ) ∈ SM is such that γ x,v ∈ E p ( M ) and v is not tangent to S p ( d ( x, p )) . Then γ p s ( x,v ) ∈ E p ( M ) for small s .If v is tangent then γ p t ( x,v ) ∈ E p ( M ) for either small t > or small t < .Proof. Suppose first that v is not tangent to S p ( d ( x, p )). Then it must bethat ddt d g ( γ x,v ( t ) , p ) (cid:12)(cid:12)(cid:12) t =0 > . The function s ddt d ( γ p s ( x,v ) ( t ) , p ) is continuous and hence(2.3) ddt d g ( γ p s ( x,v ) ( t ) , p ) (cid:12)(cid:12)(cid:12) t =0 > s . Thus γ p s ( x,v ) ∈ E p ( M ).If v is tangent to S p ( d g ( x, p )) then ddt d g ( γ x,v ( t ) , p ) (cid:12)(cid:12)(cid:12) t =0 = 0and (2.3) holds either for small positive s or for small negative s . (cid:3) Lemma 2.4.
Suppose p ∈ M and ( x, v ) ∈ SM is such that γ x,v ∈ E p ( M ) .Then γ h s ( x,v ) ∈ E p ( M ) for small s .Proof. If v is not tangent to S p ( d g ( x, p )) then proof is as for the flow p s . If v is tangent to S p ( d g ( x, p )) then γ h s ( x,v ) (0) is tangent to S p ( d g ( x, p ) + s ) or S p ( d g ( x, p ) − s ) and thus γ h s ( x,v ) ∈ E p ( M ). (cid:3) The next lemma is equation (2.2) for γ h s and γ p s . Lemma 2.5.
For all s such that γ h s ( x,v ) ∈ E p ( M ) we have d g ( γ h s ( x,v ) ( t ) , p ) ≥ ( d g ( x, p ) − s, ≤ t ≤ d g ( x, p ) ,t − d g ( x, p ) − s, t > d g ( x, p ) . For all s such that γ p s ( x,v ) ∈ E p ( M ) we have d g ( γ p s ( x,v ) ( t ) , p ) ≥ ( d g ( x, p ) , ≤ t ≤ d g ( x, p ) ,t − d g ( x, p ) , t > d g ( x, p ) . Proof.
We have for γ h s ( x,v ) by triangle inequality d g ( γ h s ( x,v ) (0) , p ) ≤ d g ( γ h s ( x,v ) (0) , x ) + d g ( x, p ) = s + d g ( x, p ) HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 7 and furthermore d g ( γ h s ( x,v ) ( t ) , γ h s ( x,v ) (0)) ≤ d g ( γ h s ( x,v ) ( t ) , p ) + d g ( γ h s ( x,v ) (0) , p ) ≤ d g ( γ h s ( x,v ) ( t ) , p ) + s + d g ( x, p ) . so t − s − d g ( x, p ) ≤ d g ( γ h s ( x,v ) ( t ) , p ) . By triangle inequality d g ( x, p ) ≤ d g ( γ h s ( x,v ) (0) , p ) + d g ( γ h s ( x,v ) (0) , x ) = d g ( γ h s ( x,v ) (0) , p ) + s. Because γ h s ( x,v ) is in E p ( M ) we get for t ≥ d g ( γ h s ( x,v ) ( t ) , p ) ≥ d g ( γ h s ( x,v ) (0) , p ) ≥ d g ( x, p ) − s. The result for γ h s ( x,v ) follows by combining these estimates. For γ p s ( x,v ) proof is similar, but we have d g ( γ p s ( x,v ) (0) , x ) = 0. (cid:3) The geodesic ray transform
As mentioned in the introduction the geodesic ray transform If : SM → R of a function f : SM → R is defined by If ( x, v ) := Z ∞−∞ f ( γ x,v ( t )) dt. Lemma 3.1.
The geodesic ray transform is well defined for f ∈ P η ( p, M ) for η > .Proof. Let ( x, v ) ∈ SM . Since If ( γ x,v ( t ) , γ ′ x,v ( t )) = If ( x, v ) for all t ∈ R ,we can assume x to be such thatmin t ∈ R d g ( γ x,v ( t ) , p ) = d g ( x, p ) . Such a point always exists on any geodesic γ since the mapping t d g ( γ ( t ) , p ) is strictly convex.By (2.1) we then have d g ( γ x,v ( t ) , p ) ≥ ( d g ( x, p ) , if | t | ≤ d g ( x, p ) , | t | − d g ( x, p ) , if 2 d g ( x, p ) < | t | . Hence for f ∈ P η ( p, M ) , η > , | If ( x, v ) | ≤ Z ∞−∞ | f ( γ x,v ( t )) | dt ≤ Z ∞−∞ C (1 + d g ( γ x,v ( t ) , p )) η dt ≤ C Z d g ( x,p )0 d g ( x, p )) η dt + Z ∞ d g ( x,p ) t − d g ( x, p )) η dt ! ≤ C (cid:18) d g ( x, p )(1 + d g ( x, p )) η + 1( η − d g ( x, p )) η − (cid:19) ≤ C ( η )(1 + d g ( x, p )) η − . (cid:3) J. LEHTONEN
Given a function f on M we define the function u f : SM → R by u f ( x, v ) = Z ∞ f ( γ x,v ( t )) dt. We observe that If ( x, v ) = u f ( x, v ) + u f ( x, − v )for all ( x, v ) ∈ SM whenever all the functions are well defined.In the next lemma we assume that f is such that If ≡ Lemma 3.2.
Suppose p ∈ M and f is a function on M such that If ≡ . (1) If f ∈ E η ( p, M ) for some η > , then | u f ( x, v ) | ≤ C ( η )(1 + d g ( x, p )) e − ηd g ( x,p ) . (2) If f ∈ P η ( p, M ) for some η > , then | u f ( x, v ) | ≤ C ( η )(1 + d ( x, p )) η − . Proof.
Since If ( x, v ) = 0 we have | u f ( x, v ) | = | u f ( x, − v ) | for all ( x, v ) ∈ SM . Thus, by Lemma 2.1, we can assume ( x, v ) to be such that γ x,v ∈E p ( M ).If f ∈ P η ( p, M ) , η >
1, using the estimate (2.2) we obtain | u f ( x, v ) | ≤ C Z d g ( x,p )0 d g ( γ x,v ( t ) , p )) η dt + Z ∞ d g ( x,p ) d g ( γ x,v ( t ) , p )) η dt ! ≤ C ( η )(1 + d g ( x, p )) η − . Similarly for f ∈ E η ( p, M ) , η >
0, we get | u f ( x, v ) | ≤ C Z d g ( x,p )0 e − ηd g ( x,p ) dt + Z ∞ d g ( x,p ) e − η ( t − d g ( x,p )) dt ! ≤ C ( η )(1 + d g ( x, p ))e − ηd g ( x,p ) . (cid:3) Next we prove that Xu f = − f , which can be seen as a reduction totransport equation. This idea is explained in details in [PSU13]. Lemma 3.3.
Suppose f ∈ P η ( p, M ) for some η > and If = 0 . Then Xu f ( x, v ) = − f ( x ) for every ( x, v ) ∈ SM .Proof. We begin by observing that X ( If ( x, v )) = Xu f ( x, v ) + X ( u f ( x, − v )) = 0so Xu f ( x, v ) = − X ( u f ( x, − v )). Hence we can assume the geodesic γ x,v tobe in E p ( M ) by Lemma 2.1. HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 9
We have Xu f ( x, v ) = dds u f ( ϕ s ( x, v )) (cid:12)(cid:12)(cid:12) s =0 = dds Z ∞ f ( γ ϕ s ( x,v ) ( t )) dt (cid:12)(cid:12)(cid:12) s =0 = Z ∞ dds f ( γ x,v ( s + t )) (cid:12)(cid:12)(cid:12) s =0 dt where the last step needs to be justified.Since we assumed our geodesic to be in E p ( M ), for t, s ≥ | dds f ( γ x,v ( t + s )) | = | d γ x,v ( t + s ) f ( γ ′ x,v ( t + s )) |≤ C (1 + d g ( γ x,v ( t + s ) , p )) η +1 ≤ C (1 + d g ( γ x,v ( t ) , p )) η +1 . Using estimate (2.1) as in the earlier proofs we obtain Z ∞ | dds f ( γ x,v ( s + t )) | dt ≤ Z ∞ C (1 + d g ( γ x,v ( t ) , p )) η +1 dt ≤ C ( η )(1 + d g ( x, p )) η , which shows that the last step earlier is justified by the dominated conver-gence theorem.Since dds f ( γ x,v ( t + s )) (cid:12)(cid:12)(cid:12) s =0 = ddt f ( γ x,v ( t ))and f ( γ x,v ( t )) → t → ∞ we have Z ∞ dds f ( γ x,v ( s + t )) (cid:12)(cid:12)(cid:12) s =0 dt = − f ( x )by the fundamental theorem of calculus. (cid:3) Regularity and decay of u f In order to prove our main theorems we need to prove C -regularity for u f given that the function f has suitable regularity and decay properties.For that we derive estimates for functions X ⊥ u f and V u f . To prove theestimates for functions X ⊥ u f and V u f we will proceed as in the case of Xu = − f (Lemma 3.3). In the proof we calculated dds f ( γ ϕ s ( x,v ) ( t )) (cid:12)(cid:12)(cid:12) s =0 = d γ ϕs ( x,v ) ( t ) f ( dds γ ϕ s ( x,v ) ( t ) (cid:12)(cid:12)(cid:12) s =0 ) . We can interpret dds γ ϕ s ( x,v ) ( t ) (cid:12)(cid:12) s =0 as a Jacobi field along the geodesic γ x,v since it is just the tangent vector field. For X ⊥ u f and V u f we proceed ina similar manner, the difference being that the geodesic flow ϕ t is replacedwith the flows h t and p t respectively. Given geodesic γ x,v we denote J γ x,v ,h ( s, t ) = ddr γ h r ( x,v ) ( t ) (cid:12)(cid:12)(cid:12) r = s and J γ x,v ,p similarly. Then J γ x,v ,h ( s, t ) is a Jacobi field along geodesic γ h s ( x,v ) for fixed s . We will write J h ( s, t ) when it is clear from the context what theundelying geodesic is. We will also use shorthand notation J h ( t ) = J h (0 , t )and J p ( t ) = J p (0 , t ).The Jacobi fields obtained in this manner turn out to be normal withinitial data (see [PU04]) J h ( s,
0) = 1 , D t J h ( s,
0) = 0 ,J p ( s,
0) = 0 , D t J p ( s,
0) = 1 . We need to have estimates for the growth of these two Jacobi fields inparticular. The first lemma giving estimates for the growth is based oncomparison theorems for Jacobi fields. See for example [Jos08, Theorem4.5.2].
Lemma 4.1.
Suppose | K ( x ) | ≤ K and γ is a geodesic. Then for Jacobifields J p and J h along a geodesic γ it holds that | J p ( t ) | ≤ C ( K ) e √ K t , | J h ( t ) | ≤ C ( K ) e √ K t , for t ≥ . This lemma tells us that these Jacobi fields will grow at most exponen-tially in presence of bounded curvature. If the curvature happens to decaysuitably we will see that these Jacobi fields will grow only at a polynomialrate.If J ( t ) is a normal Jacobi field along a geodesic γ then we can write J ( t ) = u ( t ) E ( t ) where u is a real valued function and E ( t ) is a unit normalvector field along γ . From the Jacobi equation it follows that u is a solutionto u ′′ ( t ) + K ( γ ( t )) u ( t ) = 0for t ≥ u (0) = ±| J (0) | and u ′ (0) = ±| D t J (0) | .This leads us to consider an ordinary differential equation(4.1) u ′′ ( t ) + K ( t ) u ( t ) = 0 , t ≥ ,u (0) = c ,u ′ (0) = c , for continuous K , where c , c ∈ R . Note that for J h and J p the constants c and c are either 0 or ± u is a solution to (4.1) with K such that Z ∞ t | K ( t ) | ds < ∞ HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 11 then lim t →∞ u ( t ) /t exists. We reproduce essential parts of the proof in orderto obtain a more quantitative estimate for the growth of the solution u . Lemma 4.2.
Suppose u is a solution to (4.1) with M K := Z ∞ s | K ( s ) | ds < ∞ . and c = 1 , c = 0 or other way around. Then | u ( t ) | ≤ C t + C for all t ≥ where C , C ≥ .Proof. We define A ( t ) = u ′ ( t ) and B ( t ) = u ( t ) − tu ′ ( t ) so u ( t ) = A ( t ) t + B ( t ).Fix t >
0. For all t > t it holds A ( t ) = A ( t ) − Z tt K ( s ) s (cid:18) A ( s ) + B ( s ) s (cid:19) ds,B ( t ) = B ( t ) + Z tt K ( s ) s (cid:18) A ( s ) + B ( s ) s (cid:19) ds. If we define | v ( t ) | = | A ( t ) | + | B ( t ) /t | we have | v ( t ) | ≤ | v ( t ) | + 2 Z tt s | K ( s ) || v ( s ) | ds. By a theorem of Viswanatham [Vis63] it holds | v ( t ) | ≤ ψ ( t ) on [ t , ∞ ) where ψ is a solution to ψ ′ ( t ) = 2 t | K ( t ) | ψ ( t )with ψ ( t ) = | v ( t ) | . Hence ψ ( t ) = | v ( t ) | e R tt s | K ( s ) | ds ≤ | v ( t ) | e M K and furthermore | u ( t ) | = | tv ( t ) | ≤ t e M K | v ( t ) | for t ≥ t .Then we need to estimate | v ( t ) | . In order to do so we need estimates for | u ( t ) | and | u ′ ( t ) | . We can apply Lemma 4.1 to get | u ( t ) | ≤ C ( K )e √ K t on interval [0 , t ] where we have denoted K = sup t ∈ [0 ,t ] | K ( t ) | . By inte-grating equation (4.1) we obtain | u ′ ( t ) | ≤ | u ′ (0) | + Z t | K ( t ) || u ( t ) | ds ≤ | c | + | sup t ∈ [0 ,t ] u ( t ) | K t
02 J. LEHTONEN
Thus | v ( t ) | ≤ | A ( t ) | + | B ( t ) /t | ≤ | u ( t ) /t | + 2 | u ′ ( t ) |≤ C ( K )( 1 t + 2 K t )e √ K t + 2 . By combining the estimates for intervals [0 , t ] and [ t , ∞ ) and setting t = 1 we obtain that | u ( t ) | ≤ t e M K | v (1) | + C ( K )e √ K for t ≥ (cid:3) Lemma 4.3.
Suppose | K ( x ) | ≤ K and that G is a set of geodesics suchthat M G := sup γ ∈ G Z ∞ t | K ( t ) | dt < ∞ . Let γ ∈ G . Then for Jacobi fields J p and J h along geodesic γ holds | J p ( t ) | ≤ C ( M G ) t, | J h ( t ) | ≤ C ( M G )( t + 1) . for all t ≥ . Especially the constants do not depend on the geodesic γ .Proof. Suppose geodesic γ x,v is in G . By Lemma 4.2 we obtain | J h ( t ) | ≤ C t + C , | J p ( t ) | ≤ C t + C . From the proof of that lemma we see that constants C and C above dependon the lower bound for K and the quantity Z ∞ − tK ( γ x,v ( t )) dt. Since this quantity is bounded from above by M G we can estimate constants C and C by above and get rid of the dependence on the geodesic γ x,v . Sothe constants depend only on the Gaussian curvature K and the initialconditions.Furthermore, since | J p (0) | = 0 we can drop the constant C in the estimatefor J p ( t ) by making C accordingly larger. (cid:3) Next lemma is a straightforward corollary of the preceding lemma.
Lemma 4.4.
Suppose K ∈ P η ( p, M ) for some η > . If γ ∈ E p ( M ) thenfor Jacobi fields J p and J h along geodesic γ one has | J p ( t ) | ≤ Ct, | J h ( t ) | ≤ C ( t + 1) , for all t ≥ , where the constants do not depend on the geodesic γ . HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 13
Proof.
Since K ∈ P η ( p, M ) , η >
2, we havesup γ ∈E p ( M ) Z ∞ − K ( γ ( t )) t dt < ∞ . (cid:3) With Lemmas 4.1 and 4.4 we can derive estimates for X ⊥ u f and V u f . Lemma 4.5.
Let f ∈ C ( M ) be such that If = 0 . (1) If | K ( x ) | ≤ K and f ∈ E η ( p, M ) for some η > √ K , then | X ⊥ u f ( x, v ) | ≤ C ( η, K ) e (2 √ K − η ) d g ( x,p ) for all ( x, v ) ∈ SM . (2) If f ∈ P η ( p, M ) for some η > and K ∈ P ˜ η ( p, M ) for some ˜ η > ,then | X ⊥ u f ( x, v ) | ≤ C ( η )(1 + d g ( x, p )) η − for all ( x, v ) ∈ SM .Both estimates hold also if X ⊥ is replaced by V .Proof. Let us first notice that since If = 0, it holds | X ⊥ u f ( x, − v ) | = | X ⊥ u f ( x, v ) | for all ( x, v ) ∈ SM . Thus we will assume that v is such that γ x,v ∈ E p ( M ).Firts we note that dds f ( γ h s ( x,v ) ( t )) = d γ hs ( x,v ) ( t ) f ( J h ( s, t )) . By definition X ⊥ u f ( x, v ) = dds Z ∞ f ( γ h s ( x,v ) ( t )) dt (cid:12)(cid:12)(cid:12) s =0 = Z ∞ dds f ( γ h s ( x,v ) ( t )) dt (cid:12)(cid:12)(cid:12) s =0 = Z ∞ d γ x,v ( t ) f ( J h ( s, t )) dt where the second equality holds by the dominated convergence theoremprovided that there exists function F ∈ L ([0 , ∞ )) such that(4.2) (cid:12)(cid:12)(cid:12)(cid:12) dds f ( γ h s ( x,v ) ( t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ F ( t )for all t ≥ s .Lemma 2.4 states that for small s it holds that γ h s ( x,v ) ∈ E p ( M ). Hencein the first case using Lemmas 2.5 and 4.1 we get | d γ hs ( x,v ) ( t ) f ( J h ( s, t )) | ≤ C ( K )e √ K t e − ηd g ( γ hs ( x,v ) ( t ) ,p ) ≤ ( C ( K )e ηs e √ K t e − ηd g ( x,p ) , ≤ t ≤ d g ( x, p ) ,C ( K )e ηs e √ K t e − η ( t − d g ( x,p )) , t > d g ( x, p ) , and thus Z ∞ | d γ hs ( x,v ) ( t ) f ( J h ( s, t )) | ≤ C ( η, K )e ηs e (2 √ K − η ) d g ( x,p ) . In the second case we obtain | d γ hs ( x,v ) ( t ) f ( J h ( s, t )) | ≤ ( C ( t +1)(1+ d g ( x,p ) − s ) η +1 , ≤ t ≤ d g ( x, p ) , C ( t +1)(1+ t − d g ( x,p ) − s ) η +1 , t > d g ( x, p ) . Therefore Z ∞ | d γ hs ( x,v ) ( t ) f ( J h ( s, t )) | dt ≤ C ( η )(1 − s + d g ( x, p )) η − . From these estimates we see that such a function F exists in both cases.Setting s = 0 gives the estimates for | X ⊥ u f ( x, v ) | .In case of V instead of X ⊥ we proceed in the same manner. First wenotice that | V u f ( x, − v ) | = | V u f ( x, v ) | for all ( x, v ) ∈ SM . Thus we willassume that v is such that γ x,v ∈ E p ( M ). In addition we will assume v tobe such that γ p s ( x,v ) ∈ E p ( M ) for small non-negative s , this can be done byLemma 2.3. The rest of the proof is then similar. (cid:3) From this result we see that if f is a C -function with suitable decay prop-erties then u f is in C ( SM ). Later we will approximate u f with functions u f k ∈ C ( SM ) where functions f k are compactly supported C -functions on M . The following lemma shows that functions u f k are indeed in C ( SM ). Lemma 4.6.
Suppose that f ∈ C ( M ) is compactly supported. Then u f ∈ C ( SM ) .Proof. Since f is compactly supported we have Xu f ( x, v ) = − f ( x ) ,X ⊥ u f ( x, v ) = Z ∞ d γ x,v ( t ) f ( J h ( t )) dt,V u f ( x, v ) = Z ∞ d γ x,v ( t ) f ( J p ( t )) dt. From the structural equations and the knowledge that Xu f = − f we candeduce that V Xu f , XV u f , X ⊥ Xu f , XX ⊥ u f and X u f exist.With other means we have to check that V u f , X ⊥ u f and V X ⊥ u f (orequivalently X ⊥ V u f ) exist.Let us calculate a formula for V X ⊥ u f ( x, v ) and from that we see theexistence. By definition V X ⊥ u f ( x, v ) = dds X ⊥ u f ( p s ( x, v )) (cid:12)(cid:12)(cid:12) s =0 = dds Z ∞ d γ ps ( x,v ) ( t ) f ( J γ ps ( x,v ) ,h ( t )) dt (cid:12)(cid:12)(cid:12) s =0 . We write d γ ps ( x,v ) ( t ) f ( J γ ps ( x,v ) ,h ( t )) = h∇ f ( γ p s ( x,v ) ( t )) , J γ ps ( x,v ) ,h ( t ) i . Since h D s ∇ f ( γ p s ( x,v ) ( t )) , J γ ps ( x,v ) ,h ( t ) i = Hess f ( γ p s ( x,v ) )( J p ( s, t ) , J γ ps ( x,v ) ,h ( t )) HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 15 we have dds d γ ps ( x,v ) ( t ) f ( J γ ps ( x,v ) ,h ( t )) = Hess f ( γ p s ( x,v ) )( J p ( s, t ) , J γ ps ( x,v ) ,h ( t ))+ h∇ f ( γ p s ( x,v ) )( t ) , D s J γ ps ( x,v ) ,h ( t ) i . Since Hess f and ∇ f are compactly supported we can move derivative dds into integral and deduce that V X ⊥ u f ( x, v ) exists for all ( x, v ) ∈ SM .Proofs for V u f and X ⊥ u f are once again similar. (cid:3) As a last application of Lemmas 4.1 and 4.4 we derive an estimate for thevolumes of spheres in our setting.
Lemma 4.7.
Suppose | K | ≤ K and p ∈ M . Then Vol S p ( r ) ≤ C ( K ) e √ K r . If K ∈ P η ( p, M ) for some η > , then Vol S p ( r ) ≤ Ct.
Proof.
We use polar coordinates centered at point p . Fix a tangent vec-tor v ∈ S p M and define mapping f : [0 , ∞ ) × (0 , π ) → M by f ( r, θ ) =exp p ( r e iθ v ). This gives the usual polar coordinates in which the metric g takes form g ( r, θ ) = dr + (cid:12)(cid:12)(cid:12)(cid:12) dfdθ (cid:12)(cid:12)(cid:12)(cid:12) dθ and the corresponding volume form is dV g ( r, θ ) = (cid:12)(cid:12)(cid:12)(cid:12) dfdθ (cid:12)(cid:12)(cid:12)(cid:12) dr ∧ dθ. Since exp p ( r e iθ v ) = γ p θ ( p,v ) ( r ) we have dfdθ ( r, θ ) = ddt γ p θ ( p,v ) ( r ) = J p ( r, θ )and hence the volume form on S p ( r ) is given by ι ∂ r dV g ( r, θ ) = (cid:12)(cid:12)(cid:12)(cid:12) dfdθ (cid:12)(cid:12)(cid:12)(cid:12) dθ = J p ( r, θ ) dθ. By Lemma 4.1Vol S p ( r ) ≤ Z π C ( K )e √ K r dθ = C ( K )e √ K r . In the presence of the additional assumption for the Gaussian curvautureLemma 4.4 yields Vol S p ( r ) ≤ Ct. (cid:3) Pestov identity and C -approximation In this section we prove our main theorems. The proofs are based on acertain kind of energy estimate for the operator P = V X called the Pestovidentity. We will use Pestov identity with boundary terms on submani-folds of (
M, g ). Througout this section we denote M p,r = B p ( r ) ⊂ M , asubmanifold of M with boundary S p ( r ).The following form of Pestov identity constitutes the main argument forour proofs of the main theorems. Lemma 5.1 ([IS16]) . For u ∈ C ( SM ) it holds k V Xu k L ( SM p,r ) = k XV u k L ( SM p,r ) + k Xu k L ( SM p,r ) − h KV u, V u i SM p,r − hh v, ν i V u, X ⊥ u i ∂SM p,r + hh v ⊥ , ν i V u, Xu i ∂SM p,r By using approximating sequences we can relax the regularity assump-tions for the Pestov identity. Especially the Pestov identity holds for u f with suitable f . Lemma 5.2.
Suppose either one of the following: (1) | K ( x ) | ≤ K and f ∈ E η ( p, M ) ∩ C ( M ) for some η > √ K . (2) f ∈ P η ( p, M ) ∩ C ( M ) for some η > and K ∈ P ˜ η ( p, M ) for some ˜ η > .If If = 0 , then the Pestov identity in Lemma 5.1 holds for u f .Proof. Lemmas 3.3 and 4.5 ensure that all terms of the Pestov identity arefinite.We define u k = u ϕ k f where ϕ k : M → R is a smooth cutoff function suchthat(1) 0 ≤ ϕ r ( x ) ≤ x ∈ M .(2) ϕ k ( x ) = 1 for x ∈ B p ( k ).(3) ϕ k ( x ) = 0 for x B p (2 k ).(4) |∇ ϕ | g ≤ C/k for all x ∈ M and v ∈ T x M .Such a function can be defined by ϕ k ( x ) := ϕ (cid:18) d g ( x, p ) k (cid:19) where ϕ is a suitable smooth cutoff function on R . Since functions ϕ k aresmooth and compactly supported, we have u k ∈ C ( SM ) by Lemma 4.6.Let us move on to prove the convergence. First we observe that Xu k ( x, v ) (cid:12)(cid:12) SM p,r = − f ( x )for large k . Therefore we have convergence in L -norm for the term Xu k .Next we prove convergence for XV u k under the assumption that f ∈ P η ( p, M ) ∩ C ( M ) for some η > K ∈ P ˜ η ( p, M ) for some ˜ η >
2. Firstwe notice that
XV u k = V Xu k + X ⊥ u k = X ⊥ u k HE GEODESIC RAY TRANSFORM ON TWO-DIMENSIONAL CH-MANIFOLDS 17 for large k . Similarly XV u f = X ⊥ u f so it is enough to prove that X ⊥ u k converges to X ⊥ u f . Furthermore since SM p,r has finite volume it is enoughto prove that X ⊥ u k → X ⊥ u f in L ∞ -norm.Let us denote G = { γ x,v : ( x, v ) ∈ SM p,r } . The set G fulfills the assump-tion of Lemma 4.3. Suppose ( x, v ) ∈ SM r . We have X ⊥ u k ( x, v ) − X ⊥ u f ( x, v ) = Z ∞ d γ x,v ( t ) ( ϕ k f )( J h ( t )) dt − Z ∞ d γ x,v ( t ) f ( J h ( t )) dt = Z ∞ ( ϕ k ( γ x,v ( t )) − d γ x,v ( t ) f ( J h ( t )) dt + Z ∞ f ( γ x,v ( t )) d γ x,v ( t ) ϕ k ( J h ( t )) dt. For t ≥ d g ( γ x,v ( t ) , p ) ≥ t − d g ( x, p ) ≥ t − r. Also (1 − ϕ k ( γ x,v ( t )) = 0at least for 0 ≤ t ≤ k − r and d γ x,v ( t ) ϕ k can be non-zero only in interval[ k − r, k + r ], which can be seen using triangle inequality.Hence we can estimate, with help of Lemma 4.3, that | X ⊥ u k ( x, v ) − X ⊥ u f ( x, v ) | ≤ Z ∞ k − r | d γ x,v ( t ) f ( J h ( t )) | dt + Z k + rk − r | f ( γ x,v ( t )) d γ x,v ( t ) ϕ k ( J h ( t )) | dt ≤ C Z ∞ k − r t (1 + d ( γ x,v ( t ) , p )) η +1 dt + C k Z k + rk − r t (1 + d ( γ x,v ( t ) , p )) η +1 dt ≤ C Z ∞ k − r t (1 + t − r ) η +1 dt + C k Z k + rk − r t (1 + t − r ) η +1 dt. The last two integrals do not depend on ( x, v ) and they also tend to zero as k → ∞ , which proves the L ∞ -convergence. In similar manner we can proveconvergence for V u k .Convergence for the boundary terms follows also from the L ∞ -convergencebecause the boundary ∂SM p,r has a finite volume.In the other case we proceed similarly but use Lemma 4.1 instead ofLemma 4.3. (cid:3) We are ready to prove our main theorems.
Proof of Theorem 1.
Since the geodesic ray transform is linear it is enoughto show that If = 0 implies f = 0.Let us assume f ∈ E η ( p, M ) ∩ C ( M ), η > √ K , is such that If = 0.Lemma 5.2 tell us that Pestov identity holds for u f . We will apply it onsubmanifold SM p,r .Since Xu f = − f , the term on the left hand side of the Pestov identity iszero. Because we assume Gaussian curvature to be non-positive we have −h KV u f , V u f i SM p,r ≥ . Thus if we can show that the two boundary terms tend to zero as r → ∞ ,it must be that lim r → k Xu f k L ( SM p,r ) = lim r → k f k L ( SM p,r ) = 0which proves the injectivity.Using Lemma 4.5 together with Lemma 4.7 gives (cid:12)(cid:12) hh v, ν i V u, X ⊥ u i ∂SM p,r (cid:12)(cid:12) ≤ Z ∂SM p,r | V u f || X ⊥ u f | d Σ ≤ C ( η, K ) Z ∂M p,r Z S x M e √ K − η ) d g ( x,p ) dS dV g ≤ C ( η, K ) Z ∂M p,r e √ K − η ) r dV g ≤ C ( η, K ) Z ∂M p,r e √ K − η ) r dV g ≤ C ( η, K )e √ K − η ) r Vol S p ( r ) ≤ C ( η, K )e (5 √ K − η ) r , which indeed tends to zero as r → ∞ .Similarly we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂SM p,r h v ⊥ , ν i ( V u f )( Xu f ) d Σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( η, K )e (3 √ K − η ) r . which also tends to zero as r → ∞ . (cid:3) Proof of Theorem 2.
The proof is as for the Theorem 1, just using the otherestimates provided by Lemmas 4.5 and 4.7. (cid:3)
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