Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

Arosio, Leandro; Bracci, Filippo

doi:10.1007/s13324-011-0020-3

Cited by 20 publications

(27 citation statements)

References 14 publications

(22 reference statements)

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“…However, the limit equations and can be generalized because of their simple form. As an example, let

H_{n}

be the Siegel upper half‐space

{double-struckH}_{n} = {(z_{1}, \overset{z}{true}) \in {double-struckC}^{n} | Im (z_{1}) > | \tilde{z} |^{2}},

which is biholomorphic equivalent to

{double-struckB}_{n} .

Let

- M

be an infinitesimal generator on

H_{n}

in the sense of [, Section ]. Let

M_{t}

be the solution to

\frac{{italicdM}_{t} (z)}{italicdt} = - 2 \cdot {italicDM}_{t} (z) \cdot M_{t} (z), M_{0} (z) = M (z),

where

{DM}_{t}

denotes the Jacobi matrix of

M_{t} (z)

with respect to the z ‐variables.…”

Section: Example and Remarksmentioning

confidence: 99%

“…Let

M_{t}

be the solution to

\frac{{italicdM}_{t} (z)}{italicdt} = - 2 \cdot {italicDM}_{t} (z) \cdot M_{t} (z), M_{0} (z) = M (z),

where

{DM}_{t}

denotes the Jacobi matrix of

M_{t} (z)

with respect to the z ‐variables. Provided that the solution exists and that

- M_{t}

is an infinitesimal generator on

H_{n}

for every

t \geq 0,

then

(z, t) \mapsto M_{t} (z)

is a Herglotz vector field and we can consider the Loewner equation

{\overset{g}{true}}_{t} = M_{t} (g_{t} (z)), g_{0} (z) = z

(see again [, Section ]).…”

Section: Example and Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Multiple SLE and the complex Burgers equation

Monaco

Schleißinger

2016

Mathematische Nachrichten

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In this paper we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of n slits that connect n points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.

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“…However, the limit equations and can be generalized because of their simple form. As an example, let

H_{n}

be the Siegel upper half‐space

{double-struckH}_{n} = {(z_{1}, \overset{z}{true}) \in {double-struckC}^{n} | Im (z_{1}) > | \tilde{z} |^{2}},

which is biholomorphic equivalent to

{double-struckB}_{n} .

Let

- M

be an infinitesimal generator on

H_{n}

in the sense of [, Section ]. Let

M_{t}

be the solution to

\frac{{italicdM}_{t} (z)}{italicdt} = - 2 \cdot {italicDM}_{t} (z) \cdot M_{t} (z), M_{0} (z) = M (z),

where

{DM}_{t}

denotes the Jacobi matrix of

M_{t} (z)

with respect to the z ‐variables.…”

Section: Example and Remarksmentioning

confidence: 99%

“…Let

M_{t}

be the solution to

\frac{{italicdM}_{t} (z)}{italicdt} = - 2 \cdot {italicDM}_{t} (z) \cdot M_{t} (z), M_{0} (z) = M (z),

where

{DM}_{t}

denotes the Jacobi matrix of

M_{t} (z)

with respect to the z ‐variables. Provided that the solution exists and that

- M_{t}

is an infinitesimal generator on

H_{n}

for every

t \geq 0,

then

(z, t) \mapsto M_{t} (z)

is a Herglotz vector field and we can consider the Loewner equation

{\overset{g}{true}}_{t} = M_{t} (g_{t} (z)), g_{0} (z) = z

(see again [, Section ]).…”

Section: Example and Remarksmentioning

confidence: 99%

Multiple SLE and the complex Burgers equation

Monaco

Schleißinger

2016

Mathematische Nachrichten

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“…Evolution families are well studied, with a strong link with Loewner theory and semicomplete vector fields (see for instance [10], [5] and [3]). From the properties (EF1-3) we have that (φ s,t ) 0≤s≤t are locally uniformly Lipschitz [3], i.e. for each T > 0 and r ∈ (0, 1) there exists a positive constant M(T, r) such that…”

Section: Classical Loewner Theorymentioning

confidence: 99%

“…Proposition 5. 3. Let (f t ) t≥0 be a Loewner chain of covering mappings and let be 0 ≤ s ≤ t, then the following are equivalent (1) the morphism of groups i * s,t : π 1 (Ω s ) −→ π 1 (Ω t ) is an isomorphism;…”

Section: B Bmentioning

confidence: 99%

Approximation and Loewner theory of holomorphic covering mappings

Fiacchi¹

2019

Annali di Matematica

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“…These vector fields are the natural generalizations of the function H(z, t) = −zp(z, t) in (1.1). The Loewner ODE studied in [11,10]    ∂ ∂t ϕ s,t (z) = H(ϕ s,t (z), t), a.e. t ∈ [s, ∞), z ∈ D, ϕ s,s (z) = z, s ≥ 0, z ∈ D,…”

Section: Introductionmentioning

confidence: 99%

Loewner equations on complete hyperbolic domains

Arosio

2013

Journal of Mathematical Analysis and Applications

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We prove that, on a complete hyperbolic domain $D\subset \mathbb{C}^q$, any Loewner PDE associated with a Herglotz vector field of the form $H(z,t)=\Lambda(z)+O(|z|^2)$, where the eigenvalues of $\Lambda$ have strictly negative real part, admits a solution given by a family of univalent mappings $(f_t\colon D\to \mathbb{C}^q)$ which satisfies $\cup_{t\geq 0}f_t(D)=\mathbb{C}^q$. If no real resonance occurs among the eigenvalues of $\Lambda$, then the family $(e^{\Lambda t}\circ f_t)$ is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains

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Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

Cited by 20 publications

References 14 publications

Multiple SLE and the complex Burgers equation

Multiple SLE and the complex Burgers equation

Approximation and Loewner theory of holomorphic covering mappings

Loewner equations on complete hyperbolic domains

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