MAT 638: Topics in Real Analysis: Weil-Petersson curves, traveling salesman
theorems, and minimal surfaces
Fall 2020
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
https://princeton.zoom.us/j/594605776
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
https://princeton.zoom.us/j/594605776
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
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 - math.sunysb.edu
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