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Research Presentation
Stergios M. AntonakoudisFebruary 2019
Research Presentation
Stergios M. AntonakoudisFebruary 2019
Created using LibreOffice 5 Impress with TexMaths 0.46.1 on GNU/Linux.
Research Presentation
Geometry and Dynamics in Teichmueller theory.
Research Presentation
Geometry and Dynamics in Teichmueller theory
of Riemann surfaces & their moduli spaces.
Research Presentation
“Geometry and Dynamics in Teichmueller theory”
Plan of talk:
Research Presentation
“Geometry and Dynamics in Teichmueller theory”
Plan of talk:
● Explain some context and motivation
Research Presentation
“Geometry and Dynamics in Teichmueller theory”
Plan of talk:
● Explain some context and motivation
● Research proposal and results (brief)
Research Presentation
“Geometry and Dynamics in Teichmueller theory”
Plan of talk:
● Explain some context and motivation
● Research proposal and results (brief)
For more details and reference, please see my papers and notes available at my website: https://www.dpmms.cam.ac.uk/~sa443/.
Geometry and Dynamics
Geometry and Dynamics
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces● Conformal dynamical systems
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces (their parameter spaces)
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces (their parameter spaces)
“Teichmueller theory”
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces (their parameter spaces)
“Teichmueller theory”
Today, a meeting ground ubiquitous, long history
I do research on...
Geometry and Dynamics
● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces (their parameter spaces)
“Teichmueller theory”
Today, a meeting ground ubiquitous, long history
I do research on...
…yet, still, so little is known! many problems unsolved
Is every planar Riemann surface round?
Is every planar Riemann surface round?
Is every planar Riemann surface round?
Let X open connected subset of C.
Find F: X → C injective, holomorphic with “round”
image F(X) - complement of disjoint disks and points.
Is every planar Riemann surface round?
Let X open connected subset of C.
Find F: X → C injective, holomorphic with “round”
image F(X) - complement of disjoint disks and points.
“Koebe, last century”
Is every planar Riemann surface round?
Let X open connected subset of C.
Find F: X → C injective, holomorphic with “round”
image F(X) - complement of disjoint disks and points.
“Koebe, last century”
still unsolved - thought to be intractable by most experts.
Is every planar Riemann surface round?
Let X open connected subset of C.
Find F: X → C injective, holomorphic with “round”
image F(X) - complement of disjoint disks and points.
“Koebe, last century”
no extremal problem knownto produce “round” solution
Does the map have an attracting periodic cycle?
Does the map have an attracting periodic cycle?
Answer:
Does the map have an attracting periodic cycle?
There is probably no hope that we will ever know!
Answer: ...
Does the map have an attracting periodic cycle?
There is probably no hope that we will ever know!
Answer: ...
More structure Rational map of Riemann sphere
Conformal dynamics
Does the map have an attracting periodic cycle?
Conformal dynamics
Does the map have an attracting periodic cycle?
Forward orbit of the critical point
Conformal dynamics
Does the map have an attracting periodic cycle?
Forward orbit of the critical point
If critical orbit bounded. Can you find arbitrarily close so that has an attracting periodic point?
Conformal dynamics
Does the map have an attracting periodic cycle?
Forward orbit of the critical point
If critical orbit bounded. Can you find arbitrarily close so that has an attracting periodic point?
Conformal dynamics
“Fatou, last century” Probably, the most important problem is the field!
Moduli spaces
Moduli spaces
● V. Arnold explains this concisely in the following paragraph
(from his book experimental mathematics).
Moduli spaces
● V. Arnold explains this concisely in the following paragraph
(from his book experimental mathematics).
“Poincare was interested in how to change the conditions of a problem (for instance, the boundary conditions of a differential equation), while retaining the existence and uniqueness of the solution, or how the number of solutions varies when we make some other change. Thus he started the theory of bifurcation.”.
Moduli space of Riemann surfaces
Moduli space of Riemann surfaces
● How do you parametrise all Riemann surfaces?
Moduli space of Riemann surfaces
● How do you parametrise all Riemann surfaces?
Moduli space of Riemann surfaces
● How do you parametrise all Riemann surfaces?● How do you construct and describe one closed
Riemann surface?
Moduli space of Riemann surfaces
● How do you parametrise all Riemann surfaces?● How do you construct and describe one closed
Riemann surface? ● A closed surface (connected, oriented) is described
topologically by its genus - the number of “holes”.
Moduli space of Riemann surfaces
● How do you parametrise all Riemann surfaces?● How do you construct and describe one closed
Riemann surface? ● A closed surface (connected, oriented) is described
topologically by its genus - the number of “holes”.
Here focus on closed surfaces of genus g with n points removed. We assume 2g – 2 + n is positive; E.g. genus of is at least two.
Moduli space of Riemann surfaces
Moduli space of Riemann surfaces
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces
The hyperbolic plane is identified with unit disk with the Poinare metric.
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces
The hyperbolic plane is identified with unit disk with the Poinare metric.
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces of type
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces of type
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.
Moduli space of Riemann surfaces of type
It has a natural complex structure, uniquely characterized in terms of universal property.
Moduli space of Riemann surfaces
Moduli space of Riemann surfaces
is a complex orbifold of finite volume.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
is a complex orbifold of finite volume.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
is a complex orbifold of finite volume.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
Teichmueller space of Riemann surfaces of type
is a complex orbifold of finite volume.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
Teichmueller space of Riemann surfaces of type
is a complex orbifold of finite volume.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
Teichmueller space of Riemann surfaces of type
is a complex orbifold of finite volume.
● A contractible bounded domain in
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
Teichmueller space of Riemann surfaces of type
is a complex orbifold of finite volume.
● A contractible bounded domain in ● It comes with natural geometric shape:
an intrinsic complete Finsler metric.
Moduli space of Riemann surfaces
It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.
Teichmueller space of Riemann surfaces of type
is a complex orbifold of finite volume.
● A contractible bounded domain in ● It comes with natural geometric shape:
an intrinsic complete Finsler metric.
Geometry of Teichmueller space
Geometry of Teichmueller spaceCo-tangent space at
Geometry of Teichmueller spaceCo-tangent space at
Geometry of Teichmueller spaceCo-tangent space at
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
P
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
Geometry of Teichmueller space
Teichmueller metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
Geometry of Teichmueller space
Teichmueller metric. Norm
Kobayashi metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
Geometry of Teichmueller space
Teichmueller metric. Norm
Kobayashi metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
largest norm:
Geometry of Teichmueller space
Teichmueller metric. Norm
Kobayashi metric. Norm
Co-tangent space at
Gluing oppositeparallel sides bytranslations is asurface of g=2.
P
largest norm:
Royden: Teichmueller = Kobayashi
Holomorphic rigidity
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
P
A 2x2 real matrix
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
P A(P)
AAffine map
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
P A(P)
AAffine map
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
A holomorphic totally geodesic Teichmueller disk.
P A(P)
AAffine map
Holomorphic rigidity
Theorem (SA) Let be a totally geodesic isometry. Then it is holomorphic.
In particular, it is a Teichmueller disk.
A holomorphic totally geodesic Teichmueller disk.
Teichmueller disks are abundant:
There is a holomorphic totally geodesic copy of a hyperbolic disk,
passing through every point & every direction!
P A(P)
AAffine map
Teichmuller vs symmetric domains
Teichmuller vs symmetric domains
Royden: In complex dimension two or more,
● is totally inhomogeneous.
Teichmuller vs symmetric domains
Royden: In complex dimension two or more,
● is totally inhomogeneous: (Q(X),||.||) rigidity.
Teichmuller vs symmetric domains
Royden: In complex dimension two or more,
● is totally inhomogeneous: (Q(X),||.||) rigidity.
● is discrete & equal to
Teichmuller vs symmetric domains
Royden: In complex dimension two or more,
● is totally inhomogeneous: (Q(X),||.||) rigidity.
● is discrete & equal to
Let be a Teichmueller space and a bounded symmetric domain with dimensions two or more.
Teichmuller vs symmetric domains
Theorem (SA) There are no holomorphic isometric immersions
Royden: In complex dimension two or more,
● is totally inhomogeneous: (Q(X),||.||) rigidity.
● is discrete & equal to
Let be a Teichmueller space and a bounded symmetric domain with dimensions two or more.
Teichmuller vs symmetric domains
Theorem (SA) There are no holomorphic isometric immersions
Royden: In complex dimension two or more,
● is totally inhomogeneous: (Q(X),||.||) rigidity.
● is discrete & equal to
Let be a Teichmueller space and a bounded symmetric domain with dimensions two or more.
Theorem (SA) There are no holomorphic isometric submersions
Totally geodesic subvarieties
Totally geodesic subvarieties
Classify totally geodesic subvarieties of
Moduli space and Teichmueller spaces,
of complex dimension two or more.
Basic problem:
Totally geodesic subvarieties
Classify totally geodesic subvarieties of
Moduli space and Teichmueller spaces,
of complex dimension two or more.
Basic problem:
Recent breakthroughs of “new” locii of McMullen-Mukamel-Wright!
Totally geodesic subvarieties
Classify totally geodesic subvarieties of
Moduli space and Teichmueller spaces,
of complex dimension two or more.
Basic problem:
Recent breakthroughs of “new” locii of McMullen-Mukamel-Wright!
However, I prove totally geodesic maps of classical Teichmueller spaces are always geometric (work in progress/paper in writing).
Totally geodesic subvarieties
Classify totally geodesic subvarieties of
Moduli space and Teichmueller spaces,
of complex dimension two or more.
Basic problem:
Recent breakthroughs of “new” locii of McMullen-Mukamel-Wright!
However, I prove totally geodesic maps of classical Teichmueller spaces are always geometric (work in progress/paper in writing).
My proposal: study and classify totally geodesic totally real locii.Progress: Isometries of real Teichmueller spaces are geometric (paper of this result is in writing).
Dynamics on Teichmueller space
Dynamics on Teichmueller space
Dynamics on Teichmueller space
Theorem (Denjoy-Wolff) Let be any holomorphic map. Either every orbit orbit of diverges, or the map has a fixed point .
Dynamics on Teichmueller space
Theorem (Denjoy-Wolff) Let be any holomorphic map. Either every orbit orbit of diverges, or the map has a fixed point .
Proof: Apply Schwarz’s lemma from IB complex analysis. QED
Dynamics on Teichmueller space
Dynamics on Teichmueller space
…life is more interesting!
Dynamics on Teichmueller space
The theorem of Denjoy-Wolff is true only for special domains(E.g. for bounded symmetric domains), but it fails in general -
Dynamics on Teichmueller space
The theorem of Denjoy-Wolff is true only for special domains(E.g. for bounded symmetric domains), but it fails in general - even for contractible strongly pseudo-convex domains!
Dynamics on Teichmueller space
The theorem of Denjoy-Wolff is true only for special domains(E.g. for bounded symmetric domains), but it fails in general - even for contractible strongly pseudo-convex domains!
However, we prove:
Dynamics on Teichmueller space
Theorem (SA) Let be any map that is holomorphic. Either every orbit orbit of diverges, or the map has a fixed point .
Dynamics on Teichmueller space
Theorem (SA) Let be any map that is holomorphic. Either every orbit orbit of diverges, or the map has a fixed point .
Proof: Focus on intrinsic Kobayashi metric (f is non-expanding); and a new combinatorial scheme to produce retractions.
QED
Real dynamics
Application to dynamics: entropy
Real dynamics
Application to dynamics: entropy
Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics
Real dynamics
Application to dynamics: entropy
Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics
Real dynamics
Application to dynamics: entropy
Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics
The entropy is a basic invariant of a dynamical system – here measures the growth rate of its periodic points.
Real dynamics
Application to dynamics: entropy
Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics
The entropy is a basic invariant of a dynamical system – here measures the growth rate of its periodic points.
The entropy of the family is monotone increasing in c.
Real dynamics
Application to dynamics: entropy
Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics
The entropy is a basic invariant of a dynamical system – here measures the growth rate of its periodic points.
The entropy of the family is monotone increasing in c.
Proof: (1) Conformal dynamics of quadratic rational maps of the Riemann sphere. (2) Rigidity of critically finite maps using uniqueness of Teichmueller maps.
QED
Application to dynamics: entropy
Real dynamics
Application to dynamics: entropy
Real dynamics
Application to dynamics: entropy
...requires to understandgeneral exponents
Real dynamics
Application to dynamics: entropy
Is the entropy of the family monotone increasing?
...requires to understandgeneral exponents
Real dynamics
Application to dynamics: entropy
Is the entropy of the family monotone increasing?
...requires to understandgeneral exponents
Not rational maps of the sphere.
To study the dynamics in general and prove new rigidity results -
Real dynamics
Application to dynamics: entropy
Is the entropy of the family monotone increasing?
...requires to understandgeneral exponents
Not rational maps of the sphere.
To study the dynamics in general and prove new rigidity results -
Does the flow on expand its Teichmueller metric?
Real dynamics
Application to dynamics: entropy
Is the entropy of the family monotone increasing?
...requires to understandgeneral exponents
Not rational maps of the sphere.
To study the dynamics in general and prove new rigidity results -
we need new tools and theorems from Teichmueller theory to come into play (this is work in progress).
Does the flow on expand its Teichmueller metric?
Thank you!
Thank you!
Thank you!
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