Weil-Petersson Teichmüller space

Shen, Yuliang

doi:10.1353/ajm.2018.0023

Cited by 50 publications

(55 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is convenient to start with the bounded case. See [8,27,30,33] for proofs of the results summarized in the next theorem.…”

Section: Loewner Energymentioning

confidence: 99%

“…(Here and below dz 2 denotes two-dimensional Lebesgue measure.) Moreover, a Jordan curve has finite energy if and only if it is a Weil-Petersson quasicircle, that is, its normalized welding homeomorphism belongs to the Weil-Petersson Teichmüller space [33], which appears, e.g., in the context of closed string theory and has attracted considerable interest from both mathematicians and physicists, see, e.g., [7,18,30,27]. The link with the Loewner energy goes deeper and the energy itself is intimately connected to the geometry of the Weil-Petersson Teichmüller space: it coincides with (a constant times) the universal Liouville action of Takhtajan and Teo [30], a Kähler potential for the Weil-Petersson metric.…”

Section: Introductionmentioning

confidence: 99%

“…[28,29]). An increasing homeomorphism h of R is in the Weil-Petersson class if and only if h is absolutely continuous and log h ∈ H 1/2 (R).The "only if" part of the characterization also follows easily from(2.8).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Viklund

Wang

2020

Geom. Funct. Anal.

View full text Add to dashboard Cite

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.

show abstract

“…It is convenient to start with the bounded case. See [8,27,30,33] for proofs of the results summarized in the next theorem.…”

Section: Loewner Energymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Viklund

Wang

2020

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…The functions U(f, z) and φ h (z) are important in Teichmüller theory (see [10], [15], [16]). They were used to characterize when a quasisymmetric homeomorphism is symmetric in [10] or belongs to the Weil-Petersson class in [15].…”

mentioning

confidence: 99%

“…They were used to characterize when a quasisymmetric homeomorphism is symmetric in [10] or belongs to the Weil-Petersson class in [15]. They were also used to study the BMO-Teichmüller theory in [16].…”

mentioning

confidence: 99%

Universal Teichmüller spaces and F(p,q,s) space

Feng¹,

Huo²,

Tang³

2017

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. In this paper, we introduce the F (p, s)-Teichmüller space and investigate its Schwarzian derivative model and pre-logarithmic derivative model. In particular, we prove that the pre-logarithmic derivative model is a disconnected subset of Besov type space F (p, s) and the Bers projection is holomorphic. IntroductionLet ∆ = {z : |z| < 1} be the unit disk in the complex plane C, ∆ * = C\∆ be the outside of the unit disk and S 1 = {z ∈ C : |z| = 1} be the unit circle. Let α > 0, the Bloch-type space B α consists of all holomorphic functions f on ∆ such thatand the subspace B α 0 consists of all functions f ∈ B α such thatWe denote by BMO(S 1 ) the space of all integrable functions on S 1 such thatwhere I is any arc on S 1 , |I| denotes the Lebesgue measure of I, andis the average of u over I. A holomorphic function f on ∆ belongs to BMOA(∆) if and only if it is a Poisson integral of some function which belongs to BMO(S 1 ). For any a ∈ ∆, set ϕ a (z) = z−a 1−az , z ∈ ∆. For p > 1, q > −2 and s ≥ 0, the space F (p, q, s) consists of all holomorphic functions f on the unit disk ∆ with the

show abstract

Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type

Schippers,

Staubach

2024

In the Tradition of Thurston III

View full text Add to dashboard Cite

Weil-Petersson Teichmüller space

Cited by 50 publications

References 43 publications

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Universal Teichmüller spaces and F(p,q,s) space

Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type

Contact Info

Product

Resources

About