MAT 638: Topics in Real Analysis: Weil-Petersson curves, traveling salesman theorems, and minimal surfaces

Fall 2020

Prof. Christopher Bishop
Dept of Mathematics
Stony Brook University

Office: 4-112 Mathematics Building
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631

View previous classes

        Previous lectures

Nov 17 2020 WP-talk at Princeton

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

Contact participants

        There is a contact list on the page obtained by following this page's address by "email.html".

First class Tuesday, April 25 2020.

Last class Thursday, December 3, 2020.

No class on November 24 and 26 (Thanksgiving break).

Class time - 3pm-4:20pm Tu and Th

The class was originally scheduled to be in Physics P-122, but it will now take place online. Zoom information will emailed to enrolled students. Other participants from inside or outside Stony Brook are welcome, but please contact me first to be added to the participant list, even if you are only tentative or plan to attend only occasionally.

Introduction to the course

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).

(Very) Tentative Lecture Schedule

Tuesday, Aug 25
        Course administration, rapid introduction to topic (not recorded)
Thursday, Aug 27
        Introduction: quasicircles, geometric function theory, definition of Weil-Petersson class of curves in terms of Dirichlet class.
Tuesday, Sept 1
        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.
Thursday, Sept 3
        Class canceled.
Tuesday, Sept 8
        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.
Thursday, Sept 10
        Class canceled due to internet problems (now restored).
Tuesday, Sept 15
        Previously known characterizations: Dirichlet class, Schwarzian derivatives, quasiconformal dilatations.
Thursday, Sept 17
        Conformal weldings, Shen's theorem, the Sobolev space H^{1/2}. Definition 5 on f'/|f'|.
Tuesday, Sept 22
        arclength parameterization, Mobius energy, Jones conjecture,
Thursday, Sept 24
        multi-resolution families, beta-number characterization of WP class, Discrete Jones conjecture,
Tuesday, Sept 29
        beta-numbers, Menger curvature, start biLipschitz involutions,
Thursday, Oct 1
        small involutions control betas
Tuesday, Oct 6
        BiLipschitz involutions, Smith conjecture
Thursday, Oct 8
        beta-numbers and epsilon-numbers, introduction to TST
Tuesday, Oct 13
        Proof of TST, start proof of upper bound for length - diameter
Thursday, Oct 15
        Proof of TST , finish proof of upper bound, extend to general sets
Tuesday, Oct 20
        Guest lectures: Martin Chuaqui and Yilin Wang, 30 minutes each.
        Notes by Martin Chuaqui
        Notes by Yilin Wang
Thursday, Oct 22
        Proof of TST lower bound for diameter
Tuesday, Oct 27
        Proof of TST lower bound for chord-length
Thursday, Oct 29
        Finish proof of TST for chord-length. Introduction to hyperbolic characterizations, hyperbolic space, convex hull,
Tuesday, Nov 3
        Guest lecture by Dragomir Saric:
        Saric lecture slides
Thursday, Nov 5
        Introduction to hyperbolic characterizations, hyperbolic space, convex hull, epsilon-numbers control delta-numbers.
Tuesday, Nov 10
        Curvature and the 2nd fundamental form, WP curves bound surfaces of finite total curvature. Epstein's theorem on Gauss map.
Thursday, Nov 12
        Seppi's estimate, delta-numbers control curvature WP iff renormalized area is finite
Tuesday, Nov 17
        Isoperimetric inequalities on negatively curved surfaces
Thursday, Nov 19
        The dyadic cylinder
Nov 23-27
        Thanksgiving break

Tuesday, Dec 1
        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)
Thursday, Dec 3
        Last meeting: open questions, discusssion.

Related Sildes

        Lecture slides for a talk I gave on WP curves at the Garnett-Marshall conference in Seatle, August 2019.
        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

Related Videos

Although I was aware of Takhtajan and Teo's work and David Mumford had previously posed the problem of characterizing WP curves to me, I did not start to think seriously about the problem until hearing two talks by Yilin Wang at IPAM in January of 2019
        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

Related readings

During the course of writing my paper I cited, or at least looked at, most of the following papers. Please let me know of any other related papers that you think should be shared with the class.

        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, 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: bishop - at -

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