Worm Domains are not Gromov Hyperbolic

We show that Worm domains are not Gromov hyperbolic with respect to the Kobayashi distance.

In this paper, we study the Gromov hyperbolicity of the Worm domains introduced by Diederich-Fornaess [11], which have a holomorphic annulus in the boundary and are highly non-C-convex. Worm domains play a central role in several complex variables as they provide counterexamples to several important questions. See, e.g., [17] for a review of the properties of Worm domains. We actually consider a more general class of Worms (see Definition 10), with an open Riemann surface in the boundary, and prove the following result: Theorem 1 Worms are not Gromov hyperbolic w.r.t. the Kobayashi distance.
The proof is based on Barrett's scaling (cf. [4,Sect. 4]). We rescale the Worm W obtaining in the limit a holomorphic fiber bundle, which we call a pre-Worm, with base an open hyperbolic Riemann surface and with fiber the right half-plane. We show that such a pre-Worm cannot be Gromov hyperbolic. Since the Kobayashi distance is continuous with respect to this scaling, this yields the result.

Gromov Hyperbolicity-Basic Definitions
In this section, we will review some basic definitions and properties of Gromov hyperbolic spaces. The book [8] is one of the standard references. Finally, a metric space is Gromov hyperbolic if it is δ-hyperbolic for some δ ≥ 0. Finally, recall that a metric space (X , d) is proper if closed balls are compact, and geodesic if any two points can be connected by a geodesic. A fundamental property of geodesic Gromov hyperbolic spaces is that quasigeodesics are uniformly close to geodesics, a fact which implies the following characterization of Gromov hyperbolicity.
The precise choice of a function η satisfying the above properties is completely irrelevant for what follows.
Next, given an open Riemann surface Y equipped with a smooth angle function θ : Y → R and two compact intervals I , J as above, we define We assume the following: Proof The precompactness of the domain W is a consequence of our assumption that θ −1 (J ) is compact. The domain W has defining function: We show that dr = 0 for all (z, w) ∈ ∂ W . If ∂wr = 0, this is clear. Since ∂wr = w − e iθ(z) vanishes only if w = e iθ(z) , we may assume that this identity holds. Then necessarily η(θ(z)) = 1, that is, θ(z) ∈ ∂ J , in which case by our assumption about the critical points of θ . This proves that W has smooth boundary.
Since Levi-pseudoconvexity is a local property, we may restrict the z variable to an open set U ⊂ Y where θ(z) admits a harmonic conjugate v(z), as in the proof of Proposition 5. A local defining function for the boundary of W is then given by Recalling that moduli squared (resp. real parts) of holomorphic functions are plurisubharmonic (resp. pluriharmonic), we see that e −v r is equal to a plurisubharmonic function plus e −v η • θ , which is a function of the variable z alone. If we show that the latter is subharmonic, we are done. One computes where and ∇ are the ordinary real Laplacian and gradient in C ≡ R 2 . In U , we have by Cauchy-Riemann equations. Next, notice that e −v = |e − F 2 | 2 is subharmonic. Since η and e −v are nonnegative, all we are left to do to check the nonnegativity of (e −v η•θ) is to verify that (η•θ) ≥ 0. By direct computation, we see that which is nonnegative thanks to our convexity assumption on the auxiliary function η.
Definition 10 (Worms) If the function θ is harmonic (and satisfies the assumptions on page 3), we call the domain W ⊂ Y × C a Worm.
The reader may find a picture of a Worm in Fig. 1.
For a more refined analysis, we split the boundary of W into four regions: • the spine of the Worm • the body of the Worm • the exceptional set • the caps

Remark 12
Identify the slice {w = 0} ⊂ Y × C with the Riemann surface Y . Inside Y the spine S is the closure of the domain Since the angle function θ has no critical point z ∈ θ −1 (∂ I ), the domain X in is smoothly bounded. X in is a Riemann surface contained in the boundary of the Worm W ; hence, every point of the spine S is of D'Angelo infinite type.
In what follows an important role is also played by the smoothly bounded domain Fig. 1 A Worm, whose underlying Riemann surface Y (depicted above) has genus zero and three boundary components. In this picture, the harmonic angle function θ is represented as a height function for visual clarity. In the two boxes below, one finds a generic w-slice of the worm over a point z 1 ∈ θ −1 (I ) (on the left) and z 2 ∈ θ −1 (J \ I ) (on the right). Notice that, because of the indicated choice of I and J , the surfaces X in and X out have the same topology (albeit in general different conformal structures)

Remark 13
The classical Worm domains introduced by Diederich-Fornaess [11] correspond to the case where Y = C * , θ(z) = log |z| 2 . In this case, X in is a holomorphic annulus contained in the boundary of W , of which conformal class depends on the choice of the interval I .
A"genus zero"generalization of the Diederich-Fornaess Worms is obtained choos- ] with a and b large enough, the spine S has k + 1 boundary components.

Proposition 14 The caps C and the body B consist of strongly pseudoconvex points, the exceptional set E consists of finite-type points, and the spine S consists of infinitetype points.
Proof We already remarked that S consists of infinite-type points.
In the proof of Proposition 9, we saw that the boundary of a worm has a local defining function admitting the representatioñ where ph denotes a pluriharmonic function, and that Since the latter quantity is positive on the caps (thanks to the strict convexity assumption on η), we conclude that the Worm is strictly pseudoconvex at every point of C where ∂ w is not tangent to the boundary, that is ∂ w r = 0 (or, equivalently, the vector (0, 1) is not in the complex tangent to ∂ W ). If instead ∂ w r = 0, then we have θ ∈ ∂ J (cf. the beginning of the proof of Proposition 9), and we may exploit the strong plurisubharmonicity of |e − F 2 w| 2 : Thus, every point of C is strongly pseudoconvex. We now study points (z, w) in the body B, where η • θ ≡ 0. Calculating the Levi form L (z,w)r we obtain, for (a, b) ∈ C 2 , Hence, L (z,w)r (a, b) vanishes if and only if (a, b) ∈ C 2 is a multiple of (2, wF (z)). This readily shows that the Worm is strongly pseudoconvex at every boundary point of the body where (2, wF (z)) is not in the complex tangent to the boundary. But a simple computation shows that the vector (2, wF (z)) is never complex tangent to the boundary since This shows that every point of the body B is strongly pseudoconvex.
We are left with the proof that every point of the exceptional set E is of finite type. By the Cauchy-Riemann equations, the critical points of θ are the same as the critical points of the (locally defined) holomorphic function F, and hence, they are isolated. Thus, E is a finite union of circles and circles with one point deleted (the point with w = 0, in case the circle crosses the spine). Moreover, since θ has no critical points on ∂ I , the boundary of the Worm is real-analytic in an open neighborhood of E. Thus, to verify that every point of E is of finite type, we need to check that no positive dimensional complex analytic variety lies in such a neighborhood (see, e.g., [3]). This is easy, because any point of such a variety would be of infinite type and, since we already checked that B and C consist of strongly pseudoconvex points, this would force the variety to be contained in E, which is impossible by dimension considerations (or by the open-mapping theorem).

Remark 15
The Worms are examples of domains with nontrivial, yet nicely behaved, Levi core. See [9,10], where this notion has been introduced by the second-named author and S. Mongodi. As a consequence of Proposition 14, the Levi core of a Worm is the T 1,0 bundle of its spine. A straightforward computation using [9, Proposition 4.1, part vi)] shows that the de Rham cohomology class on the spine S (or, equivalently, X in ) induced by the D'Angelo class of the Worm is represented by i(∂ − ∂)θ, which is exact if and only if the angle function θ is globally on X in the real part of a holomorphic function (that is, the pre-Worm Z (X in , θ| X in ) is trivial as a fiber bundle). This is in turn equivalent to the condition that the Diederich-Fornaess index of the Worm is 1. We refer to [9,Sect. 4] for a review of the basic theory of D'Angelo classes and to [1,9] for the implications on the Diederich-Fornaess index.
We end this section proving that Worms are complete Kobayashi hyperbolic. For this, we observe that a Worm W is naturally associated with two pre-Worms.

Definition 16 Set
where we are using the notation of Remark 12. Notice that W in ⊂ W out and W ⊂ W out .
In the remaining of the paper, if M is a complex manifold, we denote by k M its Kobayashi pseudodistance and by K M its Kobayashi-Royden pseudometric. The following lemma is proved in [14, Lemma 2.1.3].

Lemma 17
Let D ⊂ C d be a domain and k D its Kobayashi distance. If z n → ξ ∈ ∂ D and ξ admits a local holomorphic peak function, then for every neighborhood U of ξ , we get

Proposition 18 Worms are complete Kobayashi hyperbolic.
Proof Assume by contradiction that there exists a nonconvergent Cauchy sequence {x n } n in W . Passing to a subsequence, we can assume that x n → ξ ∈ ∂ W . We write x n = (z n , w n ) and ξ = (z 0 , w 0 ).
If w 0 = 0, then ξ is a pseudoconvex finite-type point by Proposition 14. By [6] (see also [18,Sect. 4]), ξ admits a local holomorphic peak function, and hence, it cannot be a Cauchy sequence by Lemma 17. Assume next that w 0 = 0, so that in particular ξ ∈ ∂ W out . Since W ⊂ W out , it follows that {x n } n is also a Cauchy sequence w.r.t. k W out , which converges to ξ ∈ ∂ W out . This contradicts the completeness of the pre-Worm W out (Proposition 8 below).

Holomorphic Fiber Bundles are not Gromov Hyperbolic
We recall a classical result from the theory of Kobayashi hyperbolic complex manifolds. If M is a complex manifold, we denote by B M ( p, r ) the k M -ball of center p and radius r . The fact that C depends only on is not stated explicitly in [16, Proposition 3.1.19], but it is clear from the (first paragraph of the) proof. We will actually use this result in the following simplified form.

Corollary 20 Let M be a Kobayashi hyperbolic complex manifold. Then there exists an absolute constant C
for all R ≥ 1 and all p ∈ M.
We introduce the following definition.

Definition 21
Let π : E → X be a holomorphic fiber bundle and z ∈ X . Then define r (z) := sup{r > 0 : the bundle trivializes over B X (z, r )} Notice that r (z) > 0 for every z ∈ X if X is Kobayashi hyperbolic. We can now prove the main result of this section. Recall [16, Theorem 3.1.9] that if X and Y are two complex manifolds then Theorem 22 Let X , F be non-compact complete Kobayashi hyperbolic complex manifolds. Let π : E → X be a holomorphic fiber bundle with fiber F and such that sup z∈X r (z) = +∞. Then (E, k E ) is not Gromov hyperbolic.
Proof We will construct a sequence {T n } n of quasigeodesic triangles in E violating the definition of Gromov hyperbolicity. Let {z n } n in X be such that r n := r (z n ) → +∞. We define and let n : B X (z n , r n /2) × F → n be a holomorphic trivialization. Let q ∈ F be any point of F. Let C ≥ 1 be the universal constant given by Corollary 20. Set t n := r n 16C . We construct the triangles in the following way. Since X and F are non-compact, for all n > 0, we can find a geodesic of X denoted γ n : [0, t n ] → X with γ n (0) = z n , and a geodesic of F denoted σ n : [0, t n ] → F with σ n (0) = q. Notice that γ n ([0, t n ]) ⊂ B X (z n , r n /8), so by Corollary 20 the curve γ n is a (C, 0)-quasigeodesic w.r.t. the Kobayashi distance of B X (z n , r n /2) (we may assume that r n ≥ 8 for every n).
By (1) the curves a n (t) = (z n , σ n (t)) and b n (t) = (γ n (t), q) are respectively a geodesic and (C, 0)-quasigeodesic of B X (z n , r n /2) × F. Moreover, a simple computation shows that the curve c n : [0, 2t n ] → B X (z n , r n /2) × F defined by is a (2C, 0)-quasigeodesic of B X (z n , r n /2) × F. Indeed, c n is a geodesic w.r.t. the distance k X + k F that is 2C-BiLipschitz to k B X (z n ,r n /2)×F in B X (z n , r n /8) × F. Hence, the triangle T n with sides a n , b n and c n is a (2C, 0)-quasigeodesic triangle in B X (z n , r n /2) × F. Notice T n is not t n -slim because k B X (z n ,r n /2)×F (c n (t n ), a n ([0, t n ]) ∪ b n ([0, t n ])) = t n . Now since n is a biholomorphism between B X (z n , r n /2) × F and n , the trianglê T n in n that is image of T n via n is again a (2C, 0) quasigeodesic triangle w.r.t. k n , and it is not t n -slim. Now the map π : E → X is non-expanding, so The triangleT n is contained in B n ( (z n , q), r n /8), and hence, it is contained in B E ( (z n , q), r n /8). By another application of Corollary 20, the distances k E and k n are C-BiLipschitz in B E ( (z n , q), r n /8), soT n are a (2C 2 , 0)-quasigeodesic triangle not (C −1 t n )-slim w.r.t. the distance k E . It follows that E is not Gromov hyperbolic.
We conclude this section highlighting an interesting class of holomorphic fiber bundles satisfying the condition sup X r = +∞.
Proposition 23 Let Y be a complex manifold and let π : E → Y be a holomorphic fiber bundle. Let X ⊂ Y be a domain. Assume that there exists a point ξ ∈ ∂ X which admits a local holomorphic peak function. Then the restricted holomorphic bundle E| X has the property sup X r = +∞. Hence, for each R > 0, we have B X (z n , R) ⊂ X ∩ U for n large enough, which implies that r (z n ) → +∞.

Corollary 24
The pre-Worms W in and W out are not Gromov hyperbolic w.r.t. its Kobayashi distance.

Proof
The domains are smoothly bounded (see Remark 12), and thus, every point in their boundary admits a local holomorphic peak function. Hence by the previous proposition the pre-Worms W in and W out satisfy sup X r = +∞ and Theorem 22 yields the result.

Barrett's Scaling and Proof of Main Theorem
In what follows, we denote by T M the holomorphic tangent bundle of a complex manifold M and by π : T M → M, the canonical projection. We denote by D ⊂ C the unit disk. Recall the following classical definition.

Proposition 26 Let M be a taut complex manifold and let {D n } n be a sequence of domains of M. Let D ∞ ⊂ M be a complete Kobayashi hyperbolic domain with simple boundary. Assume that
(i) if {x n } n is a sequence converging to x ∞ ∈ M and x n ∈ D n for all n ∈ N, then x ∞ ∈ D ∞ ; (ii) for every compact H ⊂ D ∞ , there exists N such that H ⊂ D n for n ≥ N .
Then as n → +∞ we have K D n → K D ∞ uniformly on compact subsets of T D ∞ , and k D n → k D ∞ uniformly on compact subsets of D ∞ × D ∞ .
See, e.g., [16,Chap. 5] for the notion of tautness. The idea of the proof of Proposition 26 is similar to [7,Theorem 4.3]. The proof is based on two lemmas, valid under the assumptions of the proposition.

Lemma 27
For every H ⊂ D ∞ compact and > 0, there exists N such that for all n ≥ N and for all v ∈ π −1 (H ), we have The set H is compact. Indeed, let {z n } n be a sequence in H , i.e., there exist φ n : D → D ∞ such that φ n (0) ∈ H , and |ζ n | ≤ r such that z n = φ n (ζ n ). Since φ n (0) ∈ H for all n ∈ N and D ∞ is taut (by [16,Theorem 5.1.3]), we can assume that φ n converges uniformly on compact sets to a holomorphic mapφ : D → D ∞ and that ζ n converges toζ with |ζ | ≤ r . But then z n →φ(ζ ) ∈ H . This proves that H is compact. Now, for each v ∈ π −1 (H ) let φ : D → D ∞ be such that φ(0) = π(v) and Using property (ii), there exists N such that for all n ≥ N , we have H ⊂ D n , which implies that if φ r : Finally, using the definition of the Kobayashi-Royden metric, we have

Lemma 28
For every H ⊂ D ∞ compact and > 0, there exists N such that for all n ≥ N and for all v ∈ π −1 (H ), we have Proof Fix an Hermitian metric on T D ∞ . The result immediately follows if we prove (2) for all v ∈ π −1 (H ) such that v = 1. Assume by contradiction that there exist H ⊂ D ∞ compact, > 0, and n k → +∞, v k ∈ π −1 (H ) such that v k = 1 and and hence, α k is uniformly bounded in k. We may, therefore, assume that α k converges to a limit α as k → +∞.
Since M is taut and φ k (0) ∈ H , we can assume that the sequence {φ k } k converges uniformly on compact sets to a holomorphic map φ : D → M, which satisfies the identity Finally using the definition of the Kobayashi-Royden metric, we have which is a contradiction.

Proof of Proposition 26
The uniform convergence on compact subsets of the Kobayashi-Royden metric follows from Lemmas 27 and 28. We now prove the local uniform convergence of the Kobayashi distance. In what follows, we denote by M (γ ) the Kobayashi-Royden length of a curve γ on the manifold M.
Let H ⊂ D ∞ be a compact set, and set R := diam(H ). Given p, q ∈ H and ∈ (0, 1), let γ : [0, 1] → D ∞ be a piecewise C 1 curve joining p with q and satisfying D ∞ (γ ) ≤ k D ∞ ( p, q) + . Fix o ∈ H . Then, for all t ∈ [0, 1], i.e., the support of γ is contained in B D ∞ (o, 2R + 1) which is a compact subset of D ∞ by the completeness of D ∞ . By Lemma 27, there exists N such that for all n ≥ N and for all v R) ).
In particular, for n ≥ N . For the converse, notice that by (ii) H is eventually contained in the domains D n . Given p, q ∈ H and ∈ (0, 1), let γ n : [0, 1] → D n be a piecewise C 1 curve joining p with q and satisfying D n (γ n ) ≤ k D n ( p, q) + . Fix o ∈ H and define We have that k D ∞ ( p, γ n (t n )) ≥ k D ∞ ( p, q). Indeed, this clearly holds if t n = 1. If t n < 1, then k D ∞ (o, γ n (t n )) = 2R and thus Since B D ∞ (o, 2R) is compact, by Lemma 28 there exists N such that for all n ≥ N and for all v ∈ π −1 (B D ∞ (o, 2R)), we have that K D n (v) ≥ (1 + ) −1 K D ∞ (v). Hence, where we used (3).
Let W be a Worm. We call Barrett's scaling the one-parameter group of automorphisms of Y × C given by

Lemma 30
The domain W in has simple boundary in W out .
Proof Let ϕ : D → W out be a holomorphic map such that ϕ(D) ⊂ W in . Assume that there exists ζ 0 ∈ D such that Clearly z 0 ∈ ∂ X in . If π 1 : X out ×C → X out denotes the projection to the first variable, then π 1 • φ : D → X out is a holomorphic function with image contained in X in and such that (π 1 • φ)(ζ 0 ) ∈ ∂ X in , hence by the open-mapping theorem π 1 • φ is constant. Thus, φ(D) ⊂ ∂ W in .
We are finally able to prove our main theorem. Thus, W in is Gromov hyperbolic, which contradicts Corollary 24.