STABILITY OF THE BISHOP FAMILY OF HOLOMORPHIC DISCS

WILHELM K L I N G E N B E R G ^{⋆}

⋆ Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 TÜBINGEN, Germany Email: wilhelm@moebius.mathematik.uni-tuebingen.de

1. A point p on an embedded C^{∞} -smooth real surface M ↪ ℂ^{2} is said to have a complex tangent if T_{p}M is a complex line in T_{p}ℂ^{2}. If this is the case one can choose complex coordinates z, ω = u + iv on ℂ^{2} such that T_{p}M is the z-axis and p = (0, 0). Then M is locally a graph

w = f(z)

where f is a smooth complex valued function that vanishes of second order at the origin. If ∂^{2}f/∂z∂z̄(0) ≠ 0, then a quadratic holomorphic substitution of z, w normalizes the second-order part of f at the origin to

(1)  q(z) = γ(z^{2} + z̄^{2}) + zz̄

where γ ≥ 0 is Bishop’s invariant [B]. Then we have

f = q + h + ik

1991 Mathematics Subject Classification: Primary: 58D27 Secondary: 58G20 where h, k are real functions that vanish of third order at 0. Let π : ℂ^{2} → {v = 0} ⊆ ℂ^{2}, π(z, w) = (z, u), M_{0} = πM, and assume that q : ℂ → ℝ is positive definite, i.e.

(2)  0 ≤ γ < 1/2.

In this case 0 ∈ M is called an elliptic point. By rescaling z, w we may assume that

(3)  h, k ∈ C^{∞}(Δ̄; ℝ), h, k = 0(|z|^{3}),
(4)  q + h is convex on Δ̄,
(5)  1 ≤ (q + h)(z) on ∂Δ̄.

Now M_{0} bounds the hypersurface D_{0} = {(q + h)(z) < u, v = 0} which in turn is foliated by the holomorphic discs {(q + h)(z) < t, w = t, 0 < t < 1} whose boundaries lie in M_{0} . This family of holomorphic discs survives in a possibly smaller neighborhood of 0 ∈ M by Bishop’s Theorem [B]. In this note we show that this foliated hypersurface depends smoothly on M in C^{∞}Fréchet topology and closely follow [KW]. Stability of holomorphic discs with boundary in totally real surfaces has been studied in [Be], [F], [K], [O]. It is relevant in constructions for fillable contact structures on S^{3}, see [EK], [H]. The present paper is organized as follows. In the second section we state the stability result. In the third we give smooth dependence of the Hilbert transform on the boundary in the form that it enters in the present problem. In the last section we show how the result follows from the estimate of section 3 and a normalization procedure of the real surface. The strategy follows [KW].

2. For l ∈ ℕ, l ≥ 3 let 𝔐_{l} = {w = (q + h + ik)(z); q, h, k satisfying (1) − (5) and k = 0(|z|^{l})} and let 𝔐_{0} = {w = (q + h)(z); q, h satisfying (1) − (5)} both equipped with the C^{∞}(Δ̄)Fréchet toplogy. We have π : 𝔐_{l} → 𝔐_{0}, πM = M_{0}, M_{0}^{ϵ} = M_{0} ∩ {u < ϵ}, and M_{0}^{ϵ} bounds D_{0}^{ϵ} = {(q + h)(z) < u < ϵ, v = 0}. Here and later M ∈ 𝔐_{l} is identified with its graph function f = q + h + ik. The main result of [KW] now reads as follows.

Theorem 1[KW].Let M ∈ 𝔐_{3} then there exists ϵ > 0 and a C^{∞} embedding G : (D_{0}^{ϵ}, M_{0}^{ϵ}) → (ℂ^{2}, M) with

a) G(0) = 0, dG(0) = \left[{Id \atop 0}\right] where Id : \{v = 0\} → \{v = 0\} is the identity.

b) u ∘ G(0, u) = u

c) G|w = r^{2} is holomorphic for all r^{2} < ϵ. To formulate smooth dependence of G from M we introduce more notation. Let 𝔊 → 𝔐_{3} be the locally trivial fibre bundle over 𝔐_{3} whose fibre over M ∈ 𝔐 is given by 𝔊_{M} = {G : (D_{0}^{ϵ}, M_{0}^{ϵ}) ↪ (ℂ^{2}, M); C^{∞} smooth, some ϵ > 0} and equip it with the C^{∞} Fréchet topology. Theorem 2. There exists a continuous function ϵ : 𝔐_{3} → ℝ^{+} and a C^{∞} smooth section p of 𝔊 whose values satisfy a), b), c) from Theorem 1.

3. Let q + h ∈ 𝔐_{0}, then denote by Δ_{q+h} = {z ∈ Δ : (q + h)(z) < 1}. We associate to Δ_{q+h} the diffeomorphism g(q + h) : Δ → Δ_{q+h} that maps {|z|^{2} = t} to {(q + h)(z) = t} for 0 ≤ t ≤ 1 and preserves the rays through 0. The existence and smoothness of g in z and smooth dependence on q + h follows from Morse Lemma and (5). We have g : 𝔐_{0} × (0, 1) → C^{∞}(S^{1}, ℂ) defined by g(q + h, r) = g(q + h)(·, r), where we introduced polar (0, r) coordinates on Δ , and g_{0}(q + h, r) := g(q + h, r) − g(q, r). In this regard one has the following. Proposition 3.The map g : 𝔐_{0} × (0, 1) → C^{∞}(S^{1}; ℂ) is smooth and there exists K = K(q + h, j, s) > 0 which is locally bounded in q + h such that

a) |g|_{j}, |g_{0}/r|_{j} ≤ K · r

b) |∂_{r}^{s}g|_{j}, |∂_{r}^{s}(g_{0}/r|_{j} ≤ K. Here g = g(q + h, r) ∈ C^{∞}(S^{1}; ℂ) and | · |_{j} denotes the C^{j}-sup norm. For (q + h, r) ∈ 𝔐_{0} × (0, 1), (j, α) ∈ ℕ × (0, 1) let H(q + h, r) ∈ Hom C^{j,α}(S^{1}, ℝ) be the Hilbert transform operator on the closed convex curve {(q+h)(z) = r^{2}} ⊆ Δ, parametrized by θ ∈ S^{1} . It maps ξ ∈ C^{j,α}(S^{1}, ℝ) to its harmonic conjugate with mean zero. The main technical point in the proof of Theorem 1 and 2 lies in the analysis of the asymptotic behaviour of H on the curves {(q + h)(z) = r^{2}} as r^{2} → 0. The function g, g_{0} enter in the integral operator representations of H, see (2.7) of [KW]. Proposition 4. The map H : 𝔐_{0} × (0, 1) → Hom C^{j,α}(S^{1}, ℝ) is continuous and there exists K = K(q + h, j, s) > 0, locally bounded in the first argument, such that

a) |H(q + h, r) − H(q, r)|_{j,α} ≤ K · r

b) |∂_{r}^{s}H(q + h, r)|_{j,α} ≤ K. Given Proposition 3, the above propositions follow immediately from Lemma 2.2 and Theorem 2.5 of [KW].

4. For the proof of Theorem 2 one represents G = p(m) ∈ 𝔊_{M} by the ansatz

(6)  G(z, w) = (z + zA(z, r), w + B(z, r)),

where u = r^{2}, A, B are holomorphic in z on {(q + h)(z) < r^{2}} ⊂ Δ, ReB(0, r) = 0, ImA(0, r) = 0, and one stipulates that G maps {(q + h)(z) = r^{2}, u = r^{2}} ⊂ M_{0} into M = {w = (q + h + ik)(z)}, which may be written as

(7)  r^{2} + B(z, r) = (q + h + ik)(z + zA(z, r))

for (q + h)(z) = r^{2}. This functional equation needs to be solved for A, B, these extended holomorphically to the interior of (q + h)(z) = r^{2}, and shown to depend smoothly on q + h + ik ∈ 𝔐_{3}. Using Hilbert transform one may eliminate B from (7) and is lead to the following.

(8)  F (q + h + ik, r, ξ) = 0,

where F : 𝔐_{3} ×(0, 1)×C^{j,α}(S^{1}) → C^{j,α}(S^{1}) is defined by F = (q+h)(z+zA(q+h, r)[ξ])− r^{2} + H(q + h, r) [k(z + zA(q + h, r)[ξ]]. Here,

(9)  A(q + h, r)[ξ] = ξ + iH(q + h, r)[ξ].

and (q + h)(z) = r^{2}. By Propositions 3 and 4, F is smmoth. As in [KW, §1] it follows that if ξ_{r} ∈ C^{j,α}(S^{1}), r ∈ (0, 1) satisfies (8), then

A(z, r) = A(q + h, r)[ξ]
B(z, r) = −r^{2} + (q + h + ik)(z + zA(z, r))

satisfy (7) and its holomorphic extensions in (6) will give a point G ∈ 𝔊_{M} satisfying a), b), c) of Theorem 1. Applying a Picard iteration scheme to solve (8) for ξ_{r} leads to the following existence result [KW, 3.3]:

Proposition 5.Let (j, α) ∈ ℕ × (0, 1), l ≥ 3. Then there esixts a continuous function ϵ : 𝔐 − l → ℝ^{+} and a smooth map {M × (0, ϵ(M)); M ∈ 𝔐_{3}} → C^{j,α}(S^{1}) : q + h + ik → ξ_{r}, r^{2} > ϵ^{2}(M) with

a) F (q + h + ik, r, ξr) = 0

b) |∂_{r}^{s}ξr|_{j} ≤ Kr^{l−2−s} where K = K(l, j, s, q + h + ik) > 0 is locally bounded in 𝔐_{3}. Now by extending ξ_{r} to {(q + h)(z) < r^{2}} harmonically and by (6), (10) one concludes the following as in Theorem 4.2, [KW]. Theorem 6. Let l ≥ 7, s = (l − 7)/3, then there exists a continuous section p(l) of 𝔊^{s} → 𝔐_{l} satisfying a), b), c) of Theorem 1, where 𝔊^{s} is as in the second section but with C^{s} -regularity instead of C^{∞} regularity. Now Theorem 2 follows from Theorem 6, boundary regularity of holomorphic discs in totally real surfaces, and the following Proposition 7. Let Hol(ℂ^{2}, 0) be the manifold of germs at 0 of holomorphic invertible maps of (ℂ^{2}, 0) fixing the origin, endowed with Fréchet topology. Then for any l ≥ 3 there exists a continuous map h : 𝔐_{3} → Hol (ℂ^{2}, 0), M → h_{M}, such that h_{M}(M) ∈ 𝔐_{l}.

References

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