On the Suita Conjecture
for Some Convex Ellipsoids in
It has been recently shown that for a convex domain in and the function , where is the Bergman kernel on the diagonal and the Kobayashi indicatrix, satisfies . While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute even numerically and the highest value of it obtained so far is In this paper we present precise, although rather complicated formulas for the ellipsoids (with ) and all , as well as for and on the diagonal. The Bergman kernel for those ellipsoids had been known, the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case the function is not .
For a convex domain in and the following estimates have been recently established:
is the Bergman kernel on the diagonal and
is the Kobayashi indicatrix, where denotes the unit disc. The first inequality in (1) was shown in , the proof uses -estimates for and Lempert’s theory . It is optimal, for example if is balanced with respect to (that is every intersection of with a complex line containing is a disc) then we have equality. It can be viewed as a multi-dimensional version of the Suita conjecture  proved in  (see also  for the precise characterization when equality holds).
where is a biholomorphically invariant function in . It is not clear what the optimal upper bound should be. It was in fact quite difficult to prove that one can at all have . It was done in  for ellipsoids of the form , where and . The function was also computed numerically for the ellipsoid , , based on an implicit formula for the Kobayashi function from . Our first result is the precise formula in this case:
Then for , and with , we have
For and one has
The general formula for the Kobayashi function for is known, see , but it is implicit in the sense that it requires solving a nonlinear equation which is polynomial of degree if it is an integer. It turns out however that the volume of the Kobayashi indicatrix for , that is the set where the Kobayashi function is not bigger than 1, can be found explicitly. It would be interesting to check whether Theorem 1 also holds in the non-convex case, that is when (see  for computations of the Kobayashi metric in this case).
The formula for the Bergman kernel for this ellipsoid is well known (see e.g. , Example 6.1.6):
and we can obtain the following graphs of for example for , 8, 16, 32, 64 and 128:
is a holomorphic automorphism of and therefore where attains all values of in . One can show numerically that
which was already noticed in . This is the highest value of (in arbitrary dimension) obtained so far.
In  it was also shown that for and with one has
so that in particular similarly as in Theorem 1 it is an analytic function on this part of . This raises a question whether is smooth in general. In  it was also predicted that the highest value of for convex in should be attained for for on the diagonal. The following result will answer both of these questions in the negative:
Let . Then for with we have
is on the interval but not at .
One can show that its analytic continuation to attains values below 1 and thus it follows already from (1) that cannot be analytic. To conclude that it is in fact not one has to prove much harder formula (3). Here is the full picture on the interval , the analytic continuation of from and the actual graph of :
One can check that the maximal value of for is
1. General formula for geodesics in convex complex ellipsoids
Boundary of the Kobayashi indicatrix of a convex domain at consists of the vectors where is a geodesic of satisfying . Theorems 1 and 2 will be proved using a general formula for geodesics in convex complex ellipsoids from  based on Lempert’s theory  describing geodesics of smooth strongly convex domains.
For with set
where , for , for ,
A component has a zero in if and only if . We have
For the set of vectors where forms a subset of of a full measure. The geodesics in are uniquely determined: for a given and there exists unique geodesic such that and .
2. Proof of Theorem 1
First note that the formulas for and easily follow from the first one by approximation. For and there are two possibilities for a geodesic : either crosses the axis or it does not. By and denote the respective parts of . In the first case must be of the form
parametrizes . We will need a lemma.
Let be a function of two complex variables, where and are . Then the real Jacobian of is equal to , where
The proof is left to the reader. For the mapping (10) we can compute that
3. Proof of Theorem 2
For and , where , we have by (7)
and by (8)
There are four possibilities for the set : , , , and . Denote the corresponding parts of by , , , and , respectively, so that
to get we will have to multiply the obtained volume by 2. The condition (16) transforms to
It will be convenient to substitute , , and consider the domain
One can check that in . The region may look as follows
For we will get
We may assume that , then (23) has a solution if and only if , where
and we write . This means that
We can compute that
One can check that everywhere on .
If then and using the polar coordinates in and Lemma 3 we will get
For it is more convenient to use the polar coordinates in the disk (24) instead:
the circles and intersect when , where
We can compute the second integral using the following indefinite integrals: