Dept of Mathematics

Stony Brook University

Office: 4-112 Mathematics Building

Dept. Phone: (631)-632-8290

FAX: (631)-632-7631

Lecture slides for a talk on WP curves at Differential Geometry and Geometric Analysis (DGGA) seminar, Princeton, Wed. Nov 18, 2020. The seminar link is https://princeton.zoom.us/j/594605776

In 2009 Takhtajan and Teo wrote a monograph which defined a Riemannian metric on universal Teichmuller space, which is essentially the set of quasisymmetric circle homeomorphisms or the set of planar quasicircles (we will define all this more carefully in class). Their topology has infinitely many connected components, but one of these components is exactly the closure of the smooth circle diffeomorphisms (or smooth closed curves in the plane). This component is called the Weil-Petersson class. Although Takhtajan and Teo gave several characterizations of these curve in terms of function theory (conformal maps and quasiconformal maps), it has remained an open problem to give a intrinsic geometric characterization of these curves. The purpose of this class is to discuss several such characterizations that have recently been discovered in terms of Sobolev spaces, knot energies, rates of polygonal approximations, Peter Jones' beta-numbers, Menger curvature, hyperbolic convex hulls, curvature of minimal surfaces in hyperbolic space, isoperimetric inequalities for negatively curved surfaces and renormalized area. There are further characterizations in terms of SLE (Schramm-Loewner Evolutions), the Brownian loop soup, integral geometry and operator theory, which we will mention, but not discuss in detail.

The course will generally follow a recent preprint of mine

Weil-Petersson curves, conformal energies,
beta-numbers, and minimal surfaces

and another preprint containing auxiliary results of
independent interest:

The traveling salesman theorem for Jordan curves

In particular, the latter has a self-contained proof of Peter Jones'
famous traveling salesman theorem in all finite dimensions,
and I expect to cover this during the class.

For improved readablility, I will convert the material in these preprints into a landscape format with larger print. I will follow these EUCLIDEAN LECTURE SLIDES during the Zoom meetings, and use a webcam or drawing pad for spontaneous sketches or calculations.

When proving the TST I will follow these TST LECTURE SLIDES.

For the hyperbolic conditions (starting Oct 29) I will follow these HYPERBOLIC LECTURE SLIDES.

Slides for Dec 3 (last day) OPEN PROBLEMS

I will also try to keep a list of QUESTIONS raised by the participants.

I will start with an overview of the relevant definitions and main results in the first few lectures. We will then turn to the proof, with the order and level of detail somewhat determined by the interests of the participants. Since some of the relevant topics are outside my own expertise, I may try to arrange some guest lectures as well, and participants are welcome to volunteer to present material as well, perhaps summarizing results from cited papers in analysis or geometry.

There are no problems sets or exams for this course, but enrolled students are expected to attend regularly and to actively participate with questions and discussion and ocassionally present some relevant material. We will discuss various related problems that remain open, so it is possible that such discusses could lead to new results during the course of the class.

Beside the work of Prof Takhtajan, we will also mentions results of other Stony Brook faculty including Anderson, Schul, and Sullivan.

Below I list a tentative schedule of lecture topics. This will be extended as the semester proceeds (and I hope that our meetings will be more like discussions than lectures).

Course administration, rapid introduction to topic (not recorded)

Introduction: quasicircles, geometric function theory, definition of Weil-Petersson class of curves in terms of Dirichlet class.

Statements of results: characterization of WP class in terms of Sobolev space H^{3/2}, Mobius engery and relation to knot theory, dyadic partitions of a rectifiable curve an rapid convergence of inscribed polygons, relation to Peter Jones's beta-numbers and traveling salesman theorem.

Class canceled.

Review of basic hyperbolic geomety in 2 and 3 dimensions, hyperbolic convex hull of a planar curve, Gauss and sectional curvatures, minimal surfaces, Anderson's theorem, Seppi's estimate, characterization of WP curves in terms of curvature of minimal surfaces, isomperimetric inequalities on surfaces, renormalized area.

This completes the "first introduction". The longer "second introduction" will state each of the 20 equivalent definitions of the WP class and we will prove the easy equivalences as we go along. Harder ones will be left for later.

Class canceled due to internet problems (now restored).

Previously known characterizations: Dirichlet class, Schwarzian derivatives, quasiconformal dilatations.

Conformal weldings, Shen's theorem, the Sobolev space H^{1/2}. Definition 5 on f'/|f'|.

arclength parameterization, Mobius energy, Jones conjecture,

multi-resolution families, beta-number characterization of WP class, Discrete Jones conjecture,

beta-numbers, Menger curvature, start biLipschitz involutions,

small involutions control betas

BiLipschitz involutions, Smith conjecture

beta-numbers and epsilon-numbers, introduction to TST

Proof of TST, start proof of upper bound for length - diameter

Proof of TST , finish proof of upper bound, extend to general sets

Guest lectures: Martin Chuaqui and Yilin Wang, 30 minutes each.

Notes by Martin Chuaqui

Notes by Yilin Wang

Proof of TST lower bound for diameter

Proof of TST lower bound for chord-length

Finish proof of TST for chord-length. Introduction to hyperbolic characterizations, hyperbolic space, convex hull,

Guest lecture by Dragomir Saric:

Saric lecture slides

Introduction to hyperbolic characterizations, hyperbolic space, convex hull, epsilon-numbers control delta-numbers.

Curvature and the 2nd fundamental form, WP curves bound surfaces of finite total curvature. Epstein's theorem on Gauss map.

Seppi's estimate, delta-numbers control curvature WP iff renormalized area is finite

Isoperimetric inequalities on negatively curved surfaces

The dyadic cylinder

Thanksgiving break

Two guest lectures: Rafe Mazzeo (Stanford) and David Mumford (emeritus Brown). I hope Rafe will tell us something about his work with renormalized area, and David will discuss his interest in the Weil-Petersson metric on universal Teichmuller space from a pattern recognition and computational point of view.

Mumford's slides (Open Office file)

Last meeting: open questions, discusssion.

Lecture slides for a talk on WP curves at Differential Geometry and Geometric Analysis (DGGA) seminar, Princeton, Wed. Nov 18, 2020. The seminar link is https://princeton.zoom.us/j/594605776

Lecture 1 by Wang Loewner energy via Brownian loop measure and action functional analogs of SLE/GFF couplings, Part I

Lecture 2 by Wang Loewner energy via Brownian loop measure and action functional analogs of SLE/GFF couplings, Part II

Alhfors, Conformal Invariants

Alberti, Brief introduction to geometric function theory

Alexakis and Mazzeo, 2008, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds

Almgren, 1966, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem

Almgren and Simon, 1979, Existence of embedded solutions of Plateau's problem

Ambrosio, 2015, Regularity theory for mass-minimizing currents (after Almgren-deLellis-Spadaro

Anderson, 1985, Curvature bounds for minimal surfaces in 3-manifolds

Anderson, 1983, Complete minimal hypersurfaces in hyperbolic n-manifolds

Anderson, 1982, Complete minimal varieties in hyperbolic space

Astaneh et. al., What surface minimizes entanglement entropy?

Bauer-Harms-Michor, 2011, Sobolev metrics on shape space of surfaces

Bishop, short note on the Jones conjecture

Bauer-Harms-Michor, 2014, Overview of the geometry of shape space and diffeomorphism groups

Berestycki, Introduction to the Gaussian Free Field and Liouville quantum gravity

Berestycki-Norris, lectures of Schramm-Loewner Evolution

Beurling-Ahlfors, The boundary correspondence under quasiconformal mappings

Blatt, 2019, Curves between Lipschitz and C^1 and their relation to geometric knot theory

Blatt, 2012, Boundedness and regularizing effects of O'Hara's knot energies

Blatt-Reiter, 2008, Does finite knot energy lead to differentiability?

Breuning, 2012, Immersions with bounded second fundamental form

Bruveris-Vialard, 2017, On completeness of groups of diffeomorphisms

Brylinski, 1998, The beta function of a knot

Canzani, Note for Analysis on manifolds via the Laplacian

Chang-Marshall, On a sharp inequality concerning the Dirichlet integral

Chang, 1988, Two dimensional area minimizing integral currents are classical minimal surfaces

Chavel-Feldman, 1980, Isoperimetric inequalities on curved surfaces

Chen-Cheng, 1999, Chern-Osserman inequality for minimal surfaces in H^n

Chen, 2013, Riemannian submanifolds: a survey

Martin Chuaqui, 2018, General criteria for curves to be simple.

Chuaqui-Osgood, 1994, Ahlfors-Weill extensions of conformal mappings and critical points of the Poincare map

Coskunuzer

Coskunuzer, Asymptotic Plateau problem: a survey

David-Engelstein-Mayboroda, 2018, Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one

de Oliveira Filho, 1993, Compactification of minimal submanifolds of hyperbolic space

deLellis, 2-dimensional almost area minimizing currents

Dorronsoro, Mean oscillation and Besov spaces. Canad. Math. Bull. 28 (1985), no. 4, 474–480

Epstein, The hyperbolic Gauss map and quasiconformal reflections

Federer, 1978, Colloquium lectures on geometric function theory

Federer, The singular set of area minimizing rectifiable currents with codimension one and area minimizing flat chains modulo two with arbitrary codimension

Feiszli-Jones, 2011, Curve denoising by multiscale detection and geometric shrinkage

Feiszli-Kushnarev-Leonard, 2014, Metric spaces of shapes and applications: compression, curve matching and low-dimensional representation

Feiszli-Narayan, 2015, Numerical computation of Weil-Petersson geodesics in the univesal Teichmuller space.

Fiala, 1941, Le problem des isoperimetres sur les surfaces ouvaets a courbure positive

Freedman-He-Wang, Mobius energy on knots and unknots

Gardiner and Harvey, Universal Teichmuller Space

Gardiner-Sullivan, 1992, Symmetric Structures on a Closed Curve

Garnett-Jones, BMO from dyadic BMO

Gay-Balmaz-Ratiu, 2015, The geometry of the universal Teichmuller space and the Euler-Weil-Petersson equation

Gallardo-Gutierrez,Gonzalez,Perez-Gonzalez, Pommerenke and Ratty, Locally univalent functions, VMOA and the Dirichlet space

Grafakos, Classical Fourier Analysis

Hardy, Divergent Series

Harrison-Pugh, Plateau's problem: what's next

Kushnarev, Teichons: solitonlike geodesics on universal Teichmuller space

Leonard, 2007, Efficient shape modeling: entropy, Adaptive coding, and boundary curves versus Blum's medial axis

Lin, 1989, Asymptotic behavior of area-minimizing currents in hyperbolic space

Marshall, A new proof of a sharp inequality concerning the Dirichlet integral

Michor-Mumford, 2006, Riemannian geometries on spaces of plane curves

Michor-Mumford, 2007, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

Michor-Mumford, 2013, On Euler's equation and 'EPDIFF'

Mumford, 2012, The geometry and curvature of shape spaces

Mumford-Sharon, 2006, 2-D shape analysis using conformal mappings

O'Hara, 1991, Energy of a knot

Okikiolu, 1991, Characterization of subsets of rectifiable curves in R^n

Osserman, 1978, The isometric inequality (survey in BAMS)

Pommerenke, 1978, On univalent functions, Bloch functions and VMOA

Pommerenke, 1974, On normal and automorphic functions

Radnell-Schippers-Staubach Quasiconformal maps of bordered Riemann surfaces with L^2 Beltrami differential

Radnell-Schippers-Staubach, 2017, Quasiconformal Teichmuller theory as an analytic foundation for two-dimensional conformal field theory

Schul, 2007, Subsets of rectifiable curves in Hilbert space - the analysts TSP

Seppi, 2016, Minimal disks in hyperbolic space bounded by a quasicircle at infinity

Shen, 2018, Weil-Petersson Teichmuller space

Shen-Tang-Wu, 2018, Weil-Petersson and little Teichmuller spaces on the real line

Shen-Wei 2013, Universal Teichmuller space and BMO

Shen-Wei 2014, On the tangent space to the BMO-Teichmuller space

Shen-Wu, 2019, Weil-Petersson Teichmuller space III: dependence of Riemann mappings for Weil-Petersson curves

Simons, 1968, Minimal varieties in Riemannian manifolds

DiNezza-Palatucci-Valdinoci, 2011, Hitchhiker's guide to the fractional Sobolev spaces

Solanes, 2010, Total curvature of complete surfaces in hyperbolic space

Strzelecki-Mosel, 2013, Menger curvature as a knot Energy

Tsuji, On F. Riesz;s fundamental theorem on subharmonic functions

Wang, 2019, equivalent descriptions of the Loewner energy,

White, 2018, On the compactness theorem for embedded minimal surfaces in 3-manifolds with locally bounded area and genus

white, 2009, Currents and flat chains associated to varifolds, with an application to curvature flow

White, Topics in GMT: lecture notes

White, 1987, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals

White, 2013, Minimal surfaces: lecture notes

White, 2016, Lectures on minimal surface theory

Wang, A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

Viklund and Wang, Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines

Rohde and Wang, The Loewner energy of loops and regularity of driving functions

Wang, Equivalent descriptions of Loewner energy

Wang, The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE0+

Zinsmeister, 2006 Stochastic Loewner Evolution

Send the lecturer (C. Bishop) email at:

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