On the Suita Conjecture for Some Convex Ellipsoids in ℂ2

It has been recently shown that for a convex domain Ω in C and w ∈ Ω the function FΩ(w) := ( KΩ(w)λ(IΩ(w)) )1/n , where KΩ is the Bergman kernel on the diagonal and IΩ(w) the Kobayashi indicatrix, satisfies 1 ≤ FΩ ≤ 4. While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute FΩ even numerically and the highest value of it obtained so far is 1.010182 . . . In this paper we present precise, although rather complicated formulas for the ellipsoids Ω = {|z1| + |z2| < 1} (with m ≥ 1/2) and all w, as well as for Ω = {|z1|+ |z2| < 1} and w 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 λ(IΩ(w)) is not C . Introduction For a convex domain Ω in Cn and w ∈ Ω the following estimates have been recently established: (1) 1 λ(IΩ(w)) ≤ KΩ(w) ≤ 4n λ(IΩ(w)) . Here KΩ(w) = sup{|f(w)| : f ∈ O(Ω), ∫ Ω |f |dλ ≤ 1} is the Bergman kernel on the diagonal and IΩ(w) = {φ′(0) : φ ∈ O(∆,Ω), φ(0) = w} is the Kobayashi indicatrix, where ∆ denotes the unit disc. The first inequality in (1) was shown in [3], the proof uses L2-estimates for ∂̄ and Lempert’s theory [9]. It is optimal, for example if Ω is balanced with respect to w (that is every intersection of Ω with a complex line containing w is a disc) then we have equality. It can be viewed as a multi-dimensional version of the Suita conjecture [11] proved in [2] (see also [5] for the precise characterization when equality holds). The second equality in (1) was proved in [4] using rather elementary methods. It was also shown that the constant 4 can be replaced by 16/π2 = 1.6211 . . . if Ω is in addition symmetric with respect to w. We can write (1) as 1 ≤ FΩ(w) ≤ 4, where FΩ(w) := ( KΩ(w)λ(IΩ(w)) )1/n is a biholomorphically invariant function in Ω. 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 FΩ > 1. It was done in [4] for ellipsoids of the form {|z1|+ |z2| + · · ·+ . . . |zn| < 1}, where m ≥ 1/2 and w = (b, 0, . . . , 0). The function FΩ was also computed numerically for the ellipsoid The first named author was supported by the Ideas Plus grant 0001/ID3/2014/63 of the Polish Ministry Of Science and Higher Education and the second named author by the Polish National Science Centre grant 2011/03/B/ST1/04758. 1 2 ZBIGNIEW B LOCKI, W LODZIMIERZ ZWONEK Ω = {|z1| + |z2| < 1}, m ≥ 1/2, based on an implicit formula for the Kobayashi function from [1]. Our first result is the precise formula in this case: Theorem 1. For m ≥ 1/2 define Ωm = {z ∈ C : |z1| + |z2| < 1}. Then for m 6= 2/3, m 6= 2 and b with 0 ≤ b < 1, we have λ(IΩm((b, 0))) = π 2 [ − m− 1 2m(3m− 2)(3m− 1) b − 3(m− 1) 2m(m− 2)(m+ 1) b + m 2(m− 2)(3m− 2) b + 3m 3m− 1 b − 4m− 1 2m b + m m+ 1 ] . For m = 2/3 and m = 2 one has λ(IΩ2/3((b, 0))) = π2 80 ( −65b + 40b log b+ 160b − 27b − 100b + 32 ) , λ(IΩ2((b, 0))) = π2 240 ( −3b − 25b − 120b log b+ 288b − 420b + 160 ) . The general formula for the Kobayashi function for Ωm is known, see [1], but it is implicit in the sense that it requires solving a nonlinear equation which is polynomial of degree 2m if it is an integer. It turns out however that the volume of the Kobayashi indicatrix for Ωm, 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 0 < m < 1/2 (see [10] for computations of the Kobayashi metric in this case). The formula for the Bergman kernel for this ellipsoid is well known (see e.g. [7], Example 6.1.6): KΩm(w) = 1 π2 (1− |w2|) (1/m+ 1)(1− |w2|) + (1/m− 1)|w1| ( (1− |w2|) − |w1| )3 , so that KΩm((b, 0)) = m+ 1 + (1−m)b2 π2m(1− b2)3 . 0.2 0.4 0.6 0.8 1.0 1.002 1.004 1.006 1.008 1.010 Fig. 1. FΩm((b, 0)) for Ω = {|z1| + |z2| < 1} and m = 4, 8, 16, 32, 64, 128 SUITA CONJECTURE FOR SOME CONVEX ELLIPSOIDS IN C2 3 The graphs of FΩm((b, 0)) in Figure 1 are consistent with the graphs from [4] obtained numerically using the implicit formula from [1]. Note that for t ∈ R and a ∈ ∆ the mapping Ωm 3 z 7−→ ( e (1− |a|2)1/2m (1− āz2) z1, z2 − a 1− āz2 ) is a holomorphic automorphism of Ωm and therefore FΩm((b, 0)) where 0 ≤ b < 1 attains all values of FΩm in Ωm. One can show numerically that sup m≥1/2 sup Ωm FΩm = 1.010182 . . . which was already noticed in [4]. This is the highest value of FΩ (in arbitrary dimension) obtained so far. In [4] it was also shown that for Ω = {|z1|+ |z2| < 1} and b with 0 < b < 1 one has λ(IΩ((b, 0)) = π2 6 (1− b) ( (1− b) + 8b ) , so that in particular similarly as in Theorem 1 it is an analytic function on this part of Ω. This raises a question whether λ(IΩ(w)) is smooth in general. In [4] it was also predicted that the highest value of FΩ for convex Ω in C2 should be attained for for Ω = {|z1| + |z2| < 1} on the diagonal. The following result will answer both of these questions in the negative: Theorem 2. Let Ω = {z ∈ C2 : |z1|+ |z2| < 1}. Then for b with 0 ≤ b ≤ 1/4 we have (2) λ(IΩ((b, b))) = π2 6 ( 30b − 64b + 80b − 80b + 76b − 16b − 8b + 1 ) and when 1/4 ≤ b < 1/2 (3) λ(IΩ((b, b))) = 2π2b(1− 2b)3 ( −2b3 + 3b2 − 6b+ 4 ) 3(1− b)2 + π ( 30b10 − 124b9 + 238b8 − 176b7 − 260b6 + 424b5 − 76b4 − 144b3 + 89b2 − 18b+ 1 ) 6(1− b)2 × arccos ( −1 + 4b− 1 2b2 ) + π(1− 2b) ( −180b7 + 444b6 − 554b5 + 754b4 − 1214b3 + 922b2 − 305b+ 37 ) 72(1− b) √ 4b− 1 + 4πb(1− 2b)4 ( 7b2 + 2b− 2 ) 3(1− b)2 arctan √ 4b− 1 + 4πb2(1− 2b)4(2− b) (1− b)2 arctan 1− 3b (1− b) √ 4b− 1 . The function b 7−→ λ(IΩ((b, b))) is C3 on the interval (0, 1/2) but not C3,1 at 1/4. 4 ZBIGNIEW B LOCKI, W LODZIMIERZ ZWONEK Again, the formula for the Bergman metric for this ellipsoid is known, see [6] or [7], Example 6.1.9: KΩ(w) = 2 π2 · 3(1− |w| 2)2(1 + |w|2) + 4|w1||w2|(5− 3|w|2) ( (1− |w|2)2 − 4|w1||w2| )3 ,