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Existence of geodesic spirals for the Kobayashi--Fuks metric on planar domains

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain D in C^n, can we guarantee the existence of geodesics for the Kobayashi--Fuks metric which"spiral around"in the interior of D? We find an affirmative answer to the above question for n=1 when D possesses an infinitely sheeted universal cover.


Introduction
In geometry, the concept of geodesic is basically a curve on a surface, more generally on a Riemannian manifold, representing in some sense the shortest distance between any two points on it. In a Riemannian manifold, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Given a Riemannian manifold (X, h), geodesics for h are a distinguished set of curves, which are solutions to a system of second order ODEs. These curves are uniquely specified locally by an initial point and an initial tangent direction. A geodesic, which a priori is a curve of class C 1 , is automatically of class C ∞ provided that the linear connection is of class C ∞ (see [11]). Geodesic spiral, roughly speaking, is a non-closed geodesic which is "eventually" contained inside a compact subset of the underlying manifold. Before going into a detailed study of geodesic spirals, let us first briefly recall the construction of this Kähler metric, called the Kobayashi-Fuks metric, on bounded domains in C n . For a more comprehensive study of this metric, the readers may consult the article [1].
For a bounded domain D ⊂ C n , the space for D and is given by We denote Recall that the Ricci tensors of ds 2 D are defined by and its Ricci curvature is given by [10] showed that the Ricci curvature of the Bergman metric on a bounded domain in C n is strictly bounded above by n + 1 and hence the matrix , is positive definite (see also Fuks [7]). Therefore, is a Kähler metric with Kähler potential log(K n+1 D det G D ). Since the Bergman metric is invariant under biholomorphisms it follows that so is this metric. We call this metric the Kobayashi-Fuks metric on D.
Among some major developments around the Kobayashi-Fuks metric, Dinew [4] observed that the Kobayashi-Fuks metric plays an important role in the study of the Bergman representative coordinates, a tool introduced by Bergman in his program of generalizing the Riemann mapping theorem to C n , n > 1. Dinew, in his article [5], found out a criterion that detects classes of domains which are complete under the Kobayashi-Fuks metric. In a corollary, he showed that on any bounded hyperconvex domain the Kobayashi-Fuks metric is complete, and hence in particular, by a result of Demailly [3], this metric is complete on any bounded pseudoconvex domain with Lipschitz boundary. Recently, in [1], the boundary behavior of the Kobayashi-Fuks metric has been obtained on certain classes of domains by localizing this metric and some of its related curvatures near holomorphic peak points of pseudoconvex domains. In the above article the existence of closed geodesics with a prescribed homotopy class is also discussed for the Kobayashi-Fuks metric. Since the Kobayashi-Fuks metric is closely related to the Bergman metric, it is natural to explore the classical properties that the Bergman metric enjoy in the setting of the Kobayashi-Fuks metric as well.
In this article we prove the existence of geodesic spirals for the Kobayashi-Fuks metric on smoothly bounded planar domains. This result and its proof are motivated by a theorem of Herbort [8,Theorem (3.2)] on the existence of geodesic spirals for the Bergman metric on strongly pseudoconvex domains in C n . We will use the asymptotic formula of Fefferman [6] and Boutet de Monvel and Sjöstrand [2], devised to study the asymptotic behavior of the Begman kernel, as a prime tool for our investigation concerning the geodesic spirals. As we will see in the subsequent sections of this article, the study of the boundary behavior of objects related to the Kobayashi-Fuks metric will help immensely in establishing the existence of geodesic spirals. Let us fix some notations before we state our results.
Throughout this article, "smoothly bounded" domains will mean bounded domains having C ∞ -smooth boundary. For a domain D in dimension one, we denote by ds 2 D (z) = g D (z)|dz| 2 and ds 2 D (z) =g D (z)|dz| 2 the Bergman metric and the Kobayashi-Fuks metric on D respectively. We may use the notations ds 2 D and g D (similarly, ds 2 D andg D ) interchangeably to specify the Bergman metric (similarly, the Kobayashi-Fuks metric) in dimension one. Recall that the Kähler potential of the Kobayashi-Fuks metric in dimension 1 is log A D (z) where (a) A geodesic c : R → X, which is not closed, will be called a geodesic spiral if there is a compact subset K of X and t 0 ∈ R such that c| [t 0 ,∞) ⊂ K. (b) Let c : R → X be a non-trivial geodesic and x 0 be a point in X. If there exist t 1 , t 2 ∈ R with t 1 < t 2 and c(t 1 ) = c(t 2 ) = x 0 , we will call the segment c| [t 1 ,t 2 ] a geodesic loop passing through x 0 .
Here are the main results that we present in this article: Note that every smoothly bounded domain in C has a strictly subharmonic defining function of class C ∞ . To see this consider any bounded planar domain D with smooth boundary ∂D, and fix a C ∞ defining function ρ for D which is defined on a neighborhood U of D. Without loss of generality assume U is bounded. Therefore both the following quantities Then one can check thatρ Acknowledgements. The author would like to thank D. Borah for suggesting the problem, as well as for his constant encouragement and guidance. The author is also grateful to Prof. G. Herbort for his kind inputs. Some of the results presented here, especially in Section 2, has benefited from the interactions the author had with Prof. Herbort over emails.

Boundary behavior
In this section we will use the following result on asymptotic expansion of the Bergman kernel to derive a few boundary estimates for the Kobayashi-Fuks metric.
Theorem 2.1 (Fefferman [6], Boutet de Monvel and Sjöstrand [2]). Given a bounded strongly From this point onwards D will denote a smoothly bounded domain in C, and ρ will be a strictly subharmonic defining function for D. That is - where U is a neighborhood of D and ρ : U → R is a strictly subharmonic function on U of class C ∞ with ∂ρ/∂z = 0 on the boundary ∂D = {ρ = 0}. We will simply write g(z) for g D (z) and g(z) forg D (z) since the domain under consideration is clear. Note that in dimension 1, we have and the Bergman kernel K D is given by Computing the partial derivatives of K D using formula (2.1), one obtains Using the above expressions, we compute Observe that in our case h 2 (z) > 0 on D as h 2 (z) = − ρ(z) 6 A D (z), and owing to the fact that the Bergman metric is positive definite on bounded domains. Therefore log h 2 (z) is a well defined smooth function on D.
Let us fix some further notations. We set For a smooth function f , henceforth we will write , and so on.
Proof. This result will follow from a bare-hand computation and observing carefully to what extent the logarithm terms in h 2 (z) influence the behavior of all its partial derivatives near the boundary. First note that, for any z 0 ∈ ∂D, So the function h 2 (z) stays away from zero even near the boundary ∂D. Therefore both h 2 (z) and log h 2 (z) will exhibit similar asymptotic behavior on D along with all their respective partial derivatives.
Denoting f either of the functions h 1 or h 2 , note that we have ∂ m+n f ∂z m ∂z n = C ∞ -smooth terms on D + non-smooth terms on D for m, n ∈ N ∪ {0}. We will use the following notation, ∂ m+n f ∂z m ∂z n (z) ≺ α(z) if non-smooth terms in ∂ m+n f ∂z m ∂z n (z) ≤ S α(z) near ∂D for some constant S > 0 . For example, we have and therefore we can write Furthermore, one can check ∂ 2 h 1 ∂z∂z ≺ log(−ρ), and putting all these together in (2.2) we have Now, computing successive partial derivatives of h 2 (z), one sees The lemma now follows immediately from (2.3).
To prove (ii), note that equation (2.7) implies The first term on the right hand side of the above equation is by (2.9) equal to which tends to zero as |ρ| → 0. The second term on the right hand side in (2.11) also goes to zero, faster than |ρ| log 1 |ρ| , because of the following observations we have already made: •ĝ −1 tends to zero near ∂D faster than |ρ|, • h zz is bounded by C 2 log |ρ| for some C 2 > 0, and finally, This proves (ii). Next, (iii) follows directly from (ii) with the following observation and the fact that |ρ z | is bounded on D.

Proof of the results
We need an intermediate lemma to prove Theorem 1.2: Proof. Let us consider the Lagrange function in dimension one L c, c ′ := 1 2g (c) c ′ 2 related to the Kobayashi-Fuks metric. One can write the Euler-Lagrange equation in the following complexified form To prove the expression in Lemma 3.1, we start by computing In order to compute ∂g/∂z, we will use the following relation see (2.5) and therefore obtain

Using this expression in (3.3) one gets
Using the formula which can be obtained from (3.5) and substituting the above value of ρ zz in (3.6) we arrive at Finally, substituting this expression of c ′′ in equation (3.4), we obtain the desired formula (3.1).
With Lemma 3.1 and the boundary behavior established in Section 2, we now have enough machinery to prove Theorem 1.2.
Proof of Theorem 1.2. Assume that the assertion is not true. Then for each k ∈ N there exists geodesic c k for the Kobayashi-Fuks metric satisfying Let us denote, for each k Passing to a subsequence, if necessary, we assume that the sequence of points a k converges to a point a 0 on the boundary ∂D, and the unit vectors v k converge to a unit vector v 0 ∈ C. Now, making use of (3.1), we can compute Observe that all other terms, except for 2ρ zz (a k )|v k | 2 , on the right hand side of equation (3.7) tend to zero as a k approaches the boundary point a 0 , by virtue of Lemma 2.2 and Lemma 2.3. Therefore, Because we have (ρ • c k ) ′ (0) = 0, one can check Re ρ z (a k )v k = 0. Hence the left hand side of (3.8) is a limit of non-positive real numbers, while the right hand side of (3.8) is strictly positive. This is a contradiction, which proves our theorem. Theorem 1.3 will now follow from a more general result of G. Herbort on the existence of geodesic spirals in a complete Riemannian manifold setting. Theorem 1.2, in conjuction with the following result of Herbort, immediately proves Theorem 1.3. Lemma 3.2 (Herbort). Let (X, h) be a complete Riemannian manifold which possesses an infinite sheeted universal cover. Let x 0 be a point in X such that there is no closed geodesic passing through x 0 . If there exists a compact subset K of X such that each geodesic loop passing through x 0 is contained in K, then there is a geodesic spiral passing through x 0 .
Then Theorem 1.2 implies that each geodesic loop for the Kobayashi-Fuks metric passing through z 0 must be contained inside the compact set K. To see this, let us consider any geodesic loop c| [t 1 ,t 2 ] passing through z 0 , i.e. c(t 1 ) = c(t 2 ) = z 0 . If this loop goes outside of K, then there would exist t * ∈ (t 1 , t 2 ) where the real valued function ρ • c attends a local maxima, and that c(t * ) ∈ D \ K. Therefore by the second derivative test (ρ • c) ′′ (t * ) ≤ 0, which contradicts Theorem 1.2. So we can now apply Lemma 3.2 in our set-up to conclude the existence of a geodesic spiral passing through z 0 , which proves our result.
Concluding remarks: Using similar lines of argument employed in this article, I feel one can prove the existence of geodesic spirals for the Kobayashi-Fuks metric on a smoothly bounded strongly pseudoconvex domain, carrying infinitely sheeted universal cover, in higher dimensions as well. For this one has to carefully define similar functions h 2 (z) for domains in C n and then study the asymptotic behavior of some of the partial derivatives of log h 2 (z) as was done in Section 2. Since the Kobayashi-Fuks metric is defined using the Ricci tensors of the Bergman metric, which involves computing determinant of certain (n × n) matrix and then obtaining the second order partial derivatives of that determinant, the computations become huge. So the higher dimensional case is beyond the scope of this note.