Abstract
We study questions related to critical points of the
Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Möbius invariant differential operators. In the doubly connected case the three Eis
Results & Lemmas (30)
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Lemma 2.1.
Lemma 2.1. For M = ˆΩand with canonical homology basis chosen as above, ωζ−J(ζ) = υζ−J(ζ).
Lemma 2.1. For M = ˆΩand with canonical homology basis chosen as above, ωζ−J(ζ) = υζ−J(ζ).
Theorem 3.1.
Theorem 3.1. Let Ω⊂C be a bounded finitely connected domain such that each component of C Ωconsists of a least two points. Then there exists…
Theorem 3.1. Let Ω⊂C be a bounded finitely connected domain such that each component of C \ Ωconsists of a least two points. Then there exists a compact set K ⊂Ωsuch that ∂G ∂z (z, ζ) ̸= 0, z ∈Ω\ K, ζ ∈Ω, (3.9) ∂2G ∂z∂ζ (z, ζ) ̸= 0, z, ζ ∈Ω\ K, (3.10) ∂2G
Theorem 3.2.
Theorem 3.2. Assume ∂Ωis analytic and choose a ∈Ωas above. Then the function F(z, ζ), originally defined in Ω× Ω, extends continuously to Ω×…
Theorem 3.2. Assume ∂Ωis analytic and choose a ∈Ωas above. Then the function F(z, ζ), originally defined in Ω× Ω, extends continuously to Ω× Ω, and on Ω× ∂Ωit agrees with K(z, ζ) K(a, ζ). Specifically, if (zn, ζn) ∈Ω× Ω, (z, ζ) ∈Ω× ∂Ω and (zn, ζn) →(z, ζ) as n →∞, then lim n→∞F(zn, ζn) = K(z, ζ) K(a, ζ).
Corollary 3.3.
Corollary 3.3. The limit set of the set of critical points of G(z, ζ) as ζ →∂Ωis exactly the set of zeros of K(z, ζ) for ζ ∈∂Ω. More…
Corollary 3.3. The limit set of the set of critical points of G(z, ζ) as ζ →∂Ωis exactly the set of zeros of K(z, ζ) for ζ ∈∂Ω. More precisely, for z ∈Ω, ζ ∈∂Ω, the following statements are equivalent. (i) K(z, ζ) = 0. (ii) L(z, ζ) = 0. (iii) There exist (zn, ζn) ∈Ω× Ω, (n = 1, 2, · · · ) with ∂zG(zn, ζn) = 0 such that (zn, ζn) →(z, ζ) as n →∞. (iv) For each sequence {ζn}n≥1 ⊂Ωsuch that ζn →ζ, there exist zn ∈Ωwith zn →z such that ∂zG(zn, ζn) = 0.
Theorem 4.1.
Theorem 4.1. Let Ωbe a non-compact locally compact Hausdorffspace and let Φ be a family of continuous functions Ω→[−∞, +∞]. Then there…
Theorem 4.1. Let Ωbe a non-compact locally compact Hausdorffspace and let Φ be a family of continuous functions Ω→[−∞, +∞]. Then there exists a compact topological space ΩM, unique up to homeomorphisms, such that (i) Ωis an open and dense subset of ΩM. (ii) Every f ∈Φ can be extended to a continuous function f M on ΩM. (iii) The functions f M separate points on ∂MΩ:= ΩM \ Ω. For example, when Ωis a multiply connected domain in the plane one may, for a fixed a ∈Ω, consider the family Φ of functions
Theorem 4.2.
Theorem 4.2. The gradient structure UG is precompact. To make the link with the Constantinescu-Cornea theorem we recall that as soon as a…
Theorem 4.2. The gradient structure UG is precompact. To make the link with the Constantinescu-Cornea theorem we recall that as soon as a uniform structure is introduced on a set X, there automatically arises a complete uniform space ( ¯X, ¯U) and an injection map f : X →¯X such that f(X) is dense in ¯X and U = f −1 ¯U . The triple f, ¯X, ¯U is the completion of X with respect to U. Moreover the space ¯X is compact if and only if the uniform structure U is precompact. In our setting, with
Theorem 4.3.
Theorem 4.3. The completion of the space Ωwith respect to its gradient structure UG yields a compact space ΩG = Ω∪∂GΩ, the gradient…
Theorem 4.3. The completion of the space Ωwith respect to its gradient structure UG yields a compact space ΩG = Ω∪∂GΩ, the gradient compactification of Ω. In analogy with the Martin compactification, we call the set ∂GΩ= ΩG \ Ωthe gradient boundary. In Theorem 3.2 we showed that for multiply connected domains with analytic boundary, the gradient boundary is identical with the Euclidian one. 4.2. An estimate of P. Levy. Thinking of the Martin boundary as a local concept we may, in view of Example 3
Theorem 4.4.
Theorem 4.4. In the above notation, log 1 − 2d 2R′ + d < log |z′ −ζ z −ζ | −G(z, ζ) < log 1 + 2d 2R −d for every ζ ∈Ω.
Theorem 4.4. In the above notation, log 1 − 2d 2R′ + d < log |z′ −ζ z −ζ | −G(z, ζ) < log 1 + 2d 2R −d for every ζ ∈Ω.
Proposition 5.1.
Proposition 5.1. For every a ∈Ω, the density ρ(z) of the Poincar´e metric is (5.1) ρ(z) = | ∂G ∂z (z, a)| sinh G(z, a). In particular, on…
Proposition 5.1. For every a ∈Ω, the density ρ(z) of the Poincar´e metric is (5.1) ρ(z) = | ∂G ∂z (z, a)| sinh G(z, a). In particular, on each level line G(z, a) = c, ρ(z) = − 1 2 sinh c ∂G ∂nz (z, a), i.e., ρ is proportional to the harmonic measure with respect to a, or equivalently to the Poisson kernel of the enclosed domain.
Theorem 5.2.
Theorem 5.2. The average of the Green potential Gµ, with respect to the Poincar´e metric, on a level line G(·, a) = c equals a constant…
Theorem 5.2. The average of the Green potential Gµ, with respect to the Poincar´e metric, on a level line G(·, a) = c equals a constant (independent of µ) times the total mass of µ.
Lemma 5.3.
Lemma 5.3. For n ≥1, |cn(ζ)| ≤ 1 nd(ζ, ∂Ω)n (z ∈Ω).
Lemma 5.3. For n ≥1, |cn(ζ)| ≤ 1 nd(ζ, ∂Ω)n (z ∈Ω).
Proposition 5.4.
Proposition 5.4. Let u be harmonic in some domain, u∗a harmonic conjugate of u and let ϕ > 0 be any smooth function in one real variable.…
Proposition 5.4. Let u be harmonic in some domain, u∗a harmonic conjugate of u and let ϕ > 0 be any smooth function in one real variable. Then, away from critical points of u, the level lines of u are geodesics for the metric dσ = ϕ(u∗)|∇u||dz|.
Proposition 5.5.
Proposition 5.5. The level lines of any harmonic function u are, away from crit- ical points, trajectories for the newtonian system with…
Proposition 5.5. The level lines of any harmonic function u are, away from crit- ical points, trajectories for the newtonian system with potential energy −1 2|∇u|2.
Proposition 5.6.
Proposition 5.6. The trace in configuration space Ωof the trajectories on the energy surface E = 0 of the hamiltonian system (5.11) or…
Proposition 5.6. The trace in configuration space Ωof the trajectories on the energy surface E = 0 of the hamiltonian system (5.11) or (5.10) are exactly the inverse images under the conformal map w = f(z) = u + iu∗of the straight lines in the w-plane. As an application of the above we may take u to be the Green’s function of a domain Ω⊂C: u(z) = G(z, a), a ∈Ωfixed. With Ω′ = Ω\{z1(a), . . . , zg(a)}, where z1(a), . . . , zg(a) are the critical points of G(z, a), we conclude from Proposition 5.4 t
Theorem 6.1.
Theorem 6.1. With ρ defined by (6.26) we have (i) 1 < ρ < √ R. (ii) The Green’s function G(z, a) of A1,R has, for any given a ∈A1,R, a…
Theorem 6.1. With ρ defined by (6.26) we have (i) 1 < ρ < √ R. (ii) The Green’s function G(z, a) of A1,R has, for any given a ∈A1,R, a unique critical point z = zG(a). This is located on the same diameter as a but on the opposite side of the hole. More precisely, zG(a) = −g(|a|) a |a|, where g : (1, R) →(1, R) is an increasing function which maps (1, R) onto the relatively compact subinterval (ρ, R/ρ). It satisfies g(x)g(R/x) = R for 1 < x < R, in particular g( √ R) = √
Corollary 6.2.
Corollary 6.2. The annulus A1,R is the disjoint union of the set of critical points zG(a) of the Green’s function, the set of zeros zK(a)…
Corollary 6.2. The annulus A1,R is the disjoint union of the set of critical points {zG(a)} of the Green’s function, the set of zeros {zK(a)} of the Bergman kernel and the two circles {|z| = ρ} and {|z| = R ρ }.
Corollary 6.3.
Corollary 6.3. Let R > 1. There exist a sequence ρn, 1 = ρ0 < ρ1 < ρ2 < · · · < √ R with limn→∞ρn = √ R such that, on setting An = z ∈C: ρn…
Corollary 6.3. Let R > 1. There exist a sequence {ρn}, 1 = ρ0 < ρ1 < ρ2 < · · · < √ R with limn→∞ρn = √ R such that, on setting An = {z ∈C : ρn < |z| < R/ρn}, the critical points for the Green’s function of An are all contained in An+1. One might ask what would be corresponding statement in higher connectivity. 6.4. Spectral point of view. The Green’s function can also be obtained via spec- tral problems for the Laplace operator. This requires a choice of a metric, or at least a volume form. For
Lemma 6.4.
Lemma 6.4. Let τ ∈C, Im τ > 0. Let u, v ∈R Z be fixed and define for Re s > 1 the zeta function ζτ(s; u, v) = X (m,n)∈Z2 (0,0) Im τ |mτ +…
Lemma 6.4. Let τ ∈C, Im τ > 0. Let u, v ∈R\ Z be fixed and define for Re s > 1 the zeta function ζτ(s; u, v) = X (m,n)∈Z2\{(0,0)} Im τ |mτ + n|2 s e2πi(mu+nv). Then the analytic continuation at s = 1 of ζτ(s; u, v) is ζτ(1; u, v) = 2π2v2Im τ −2π log |ϑ1(u −vτ; τ) η(τ) |,
Lemma 6.5.
Lemma 6.5. The Epstein zeta function ζτ(s) has a meromorphic continuation to all C with a simple pole at s = 1, given by ζτ(s) = π s −1 +…
Lemma 6.5. The Epstein zeta function ζτ(s) has a meromorphic continuation to all C with a simple pole at s = 1, given by ζτ(s) = π s −1 + 2π γ −ln(2 √ Im τ|η(τ)|2) + O(s −1) where γ is the Euler constant. In particular ζτ(0) = −1. Continuing the computation of the Green’s function we have, using Lemma 6.4, G(z, w) = lim s→1 Gs(z, w)
Lemma 7.1.
Lemma 7.1. [6] Let z = f(t) = at+b ct+d, ad −bc = 1, λ(t) = (ct + d)−1, so that f ′(t) = λ(t)2. Then, for any smooth function F(z) and any…
Lemma 7.1. [6] Let z = f(t) = at+b ct+d, ad −bc = 1, λ(t) = (ct + d)−1, so that f ′(t) = λ(t)2. Then, for any smooth function F(z) and any positive integer m, ∂m ∂tm (F(f(t))λ(t)1−m) = ∂mF ∂zm (f(t))λ(t)1+m. For m = 1 this is the ordinary chain rule, holding for any change of coordinate f, whereas for m ≥2 the formula holds only for M¨obius transformations. In any projective coordinate t, a natural fundamental set (basis) of solutions of the equation Λmu = 0 is {1, t, . . . , tm−1}. Considering
Proposition 7.2.
Proposition 7.2. In the notation of Section 5.2, plus (3.4), the quantities (7.20) p(ζ) = −c0(ζ), (7.21) r(ζ) = −2c1(ζ) = 2∂p ∂z, (7.22)…
Proposition 7.2. In the notation of Section 5.2, plus (3.4), the quantities (7.20) p(ζ) = −c0(ζ), (7.21) r(ζ) = −2c1(ζ) = 2∂p ∂z, (7.22) q(ζ) = −6(∂c1(ζ) ∂ζ −2c2(ζ)) = 6πℓ(ζ, ζ) transform under conformal mapping as, respectively, the real part of a 0-connection, an affine connection and a projective connection.
Proposition 8.1.
Proposition 8.1. Under a conformal mapping f: z 7→w, the curvature of a curve Γ and its image curve f(Γ) are related by (8.3) Im ( z, w…
Proposition 8.1. Under a conformal mapping f : z 7→w, the curvature of a curve Γ and its image curve f(Γ) are related by (8.3) Im ({z, w}1dw) = κzdsz −κwdsw. Here dsz = |dz|, dsw = |dw|. Example 8.1. (The curvature in terms of a real parameter.) With Γ = R and z = f(w), so that z = f(t) with t = Re w parametrizes f(Γ), we have κw = 0, which gives the well-known formula κz = Im ( d dw (log dz dw ) dw dsz ) + κw dsw
Proposition 8.2.
Proposition 8.2. The curvature κ = κ(z, dz ds) of the geodesic passing through a point z and having direction dz ds (a unit vector) is…
Proposition 8.2. The curvature κ = κ(z, dz ds) of the geodesic passing through a point z and having direction dz ds (a unit vector) is given by (8.8) κ(z, dz ds) = −Im (r(z)dz ds) = −Im (2∂p ∂z dz ds) = −∂p ∂n, where ∂p ∂n denotes the derivative in the rightward normal direction of the curve. In particular, the sharp bound
Corollary 8.3.
Corollary 8.3. For a simply connected domain Ωprovided with its Poincar´e met- ric, the curvature for any geodesics through a point z is…
Corollary 8.3. For a simply connected domain Ωprovided with its Poincar´e met- ric, the curvature for any geodesics through a point z is subject to the estimate |κ(z, dz ds)| ≤ 2 d(z, ∂Ω). It is allowed here that Ω⊂P contains the point of infinity, and at least in this generality the corollary is sharp. Indeed, with Ω= P\D(0, ǫ), ǫ > 0 small, the circle with center at any finite point c ∈Ω, and having radius |c|, is almost a geodesic for the Poincar´e metric in Ω. The curvature for this circle is
Proposition 9.1.
Proposition 9.1. The reproducing kernels for H(Ω) and He(Ω) are, respectively, k(z, ζ) = 1 2π (Na(z, ζ) −G(z, ζ)), ke(z, ζ) = 1 2π (Na(z,…
Proposition 9.1. The reproducing kernels for H(Ω) and He(Ω) are, respectively, k(z, ζ) = 1 2π (Na(z, ζ) −G(z, ζ)), ke(z, ζ) = 1 2π (Na(z, ζ) −Gγ(z, ζ)). In other words, k(·, ζ) ∈H(Ω) and (9.13) u(ζ) = D(u, k(·, ζ)) (u ∈H(Ω)), and similarly for ke(·, ζ).
Proposition 9.2.
Proposition 9.2. The reduced Bergman kernel, i.e., the reproducing kernel for Be(Ω), is given by Ke(z, ζ) = 2 π ∂2Na(z, ζ) ∂z∂¯ζ = −2 π…
Proposition 9.2. The reduced Bergman kernel, i.e., the reproducing kernel for Be(Ω), is given by Ke(z, ζ) = 2 π ∂2Na(z, ζ) ∂z∂¯ζ = −2 π ∂2Gγ(z, ζ) ∂z∂¯ζ = ∂2K(z, ζ) ∂z∂¯ζ for any choices of a and γ as above. Similarly, the corresponding adjoint kernel is Le(z, ζ) = 2 π
Proposition 10.1.
Proposition 10.1. With F, G (regarded as 1−m 2 -forms) and g (regarded as a 1+m 2 -form) analytic in Ω, smooth up to ∂Ω, we have (10.2) Z Ω…
Proposition 10.1. With F, G (regarded as 1−m 2 -forms) and g (regarded as a 1+m 2 -form) analytic in Ω, smooth up to ∂Ω, we have (10.2) Z Ω (ΛmF)¯gωm−1dzd¯z = im−1(m −1)! Z ∂Ω F ¯g(dz) 1−m 2 (d¯z) 1+m
Corollary 10.2.
Corollary 10.2. For m = 0, 1, 2,..., (10.6) (F, G)−m = (ΛmF, ΛmG)m, hence (F, F)−m ≥0 with equality if and only if ΛmF = 0.
Corollary 10.2. For m = 0, 1, 2, . . ., (10.6) (F, G)−m = (ΛmF, ΛmG)m, hence (F, F)−m ≥0 with equality if and only if ΛmF = 0.
Proposition 10.3.
Proposition 10.3. Bm,e(Ω)⊥consists of those elements in Bm(Ω) which extend to the Schottky double ˆΩas holomorphic differentials of order…
Proposition 10.3. Bm,e(Ω)⊥consists of those elements in Bm(Ω) which extend to the Schottky double ˆΩas holomorphic differentials of order 1+m 2 .
Theorem 11.1.
Theorem 11.1. Let Km(z, ζ) denote the reproducing kernel for Am(Ω) and Lm(z, ζ) the adjoint kernel. Then, (11.1) Λm¯ΛmKm(z, ζ) = Km,e(z,…
Theorem 11.1. Let Km(z, ζ) denote the reproducing kernel for Am(Ω) and Lm(z, ζ) the adjoint kernel. Then, (11.1) Λm¯ΛmKm(z, ζ) = Km,e(z, ζ), (11.2) ΛmΛmLm(z, ζ) = Lm,e(z, ζ), where the first Λm acts on the z-variable and the second one on ζ. The leading term in the singularity of Lm is given by Lm(z, ζ) = − im−1 πm!(m −1)!(z −ζ)m−1 log(z −ζ) + less singular terms.
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