research presentation - university of cambridgesa443/research-slides.pdf · research presentation...

Research Presentation

Stergios M. AntonakoudisFebruary 2019


Stergios M. AntonakoudisFebruary 2019

Created using LibreOffice 5 Impress with TexMaths 0.46.1 on GNU/Linux.


Geometry and Dynamics in Teichmueller theory.


Geometry and Dynamics in Teichmueller theory

of Riemann surfaces & their moduli spaces.


“Geometry and Dynamics in Teichmueller theory”

Plan of talk:



Plan of talk:

● Explain some context and motivation



Plan of talk:


● Research proposal and results (brief)



Plan of talk:


● 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/.

https://www.dpmms.cam.ac.uk/~sa443/

Geometry and Dynamics


I do research on...


● The geometry of Riemann surfaces

I do research on...


● The geometry of Riemann surfaces● Conformal dynamical systems

I do research on...


● The geometry of Riemann surfaces,● Conformal dynamical systems and their

I do research on...


● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces

I do research on...


● The geometry of Riemann surfaces,● Conformal dynamical systems and their● Moduli spaces (their parameter spaces)

I do research on...



“Teichmueller theory”

I do research on...




Today, a meeting ground ubiquitous, long history

I do research on...




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?


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.






no extremal problem knownto produce “round” solution

Does the map have an attracting periodic cycle?


Answer:


There is probably no hope that we will ever know!

Answer: ...


There is probably no hope that we will ever know!

Answer: ...

More structure Rational map of Riemann sphere

Conformal dynamics


Conformal dynamics


Forward orbit of the critical point

Conformal dynamics



If critical orbit bounded. Can you find arbitrarily close so that has an attracting periodic point?

Conformal dynamics



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


● How do you parametrise all Riemann surfaces?


● How do you parametrise all Riemann surfaces?● How do you construct and describe one closed

Riemann surface?



Riemann surface? ● A closed surface (connected, oriented) is described

topologically by its genus - the number of “holes”.



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.


It is a discrete subgroup of holomorphic automorphismsof the unit disk in the plane.


The hyperbolic plane is identified with unit disk with the Poinare metric.




Moduli space of Riemann surfaces of type



Moduli space of Riemann surfaces of type

It has a natural complex structure, uniquely characterized in terms of universal property.


is a complex orbifold of finite volume.


It is a discrete subgroup of holomorphic automorphismsof bounded complex domain.




Teichmueller space of Riemann surfaces of type






● A contractible bounded domain in





● 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


Teichmueller metric. Norm

Co-tangent space at



Co-tangent space at

P



Co-tangent space at

Gluing oppositeparallel sides bytranslations is asurface of g=2.

P



Kobayashi metric. Norm

Co-tangent space at


P




Co-tangent space at


P

largest norm:




Co-tangent space at


P

largest norm:

Royden: Teichmueller = Kobayashi

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




P A(P)

AAffine map




A holomorphic totally geodesic Teichmueller disk.

P A(P)

AAffine map




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


Royden: In complex dimension two or more,

● is totally inhomogeneous.



● 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 immersions






Theorem (SA) There are no holomorphic isometric immersions





Theorem (SA) There are no holomorphic isometric submersions

Totally geodesic subvarieties


Classify totally geodesic subvarieties of

Moduli space and Teichmueller spaces,

of complex dimension two or more.

Basic problem:





Basic problem:

Recent breakthroughs of “new” locii of McMullen-Mukamel-Wright!





Basic problem:


However, I prove totally geodesic maps of classical Teichmueller spaces are always geometric (work in progress/paper in writing).





Basic problem:


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


Theorem (Denjoy-Wolff) Let be any holomorphic map. Either every orbit orbit of diverges, or the map has a fixed point .


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


…life is more interesting!


The theorem of Denjoy-Wolff is true only for special domains(E.g. for bounded symmetric domains), but it fails in general -


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!


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:


Theorem (SA) Let be any map that is holomorphic. Either every orbit orbit of diverges, or the map has a fixed point .


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


Theorem (Milnor-Thurston) Bifurcations appear monotonically within the real family of real quadratics

Real dynamics



The entropy is a basic invariant of a dynamical system – here measures the growth rate of its periodic points.

Real dynamics




The entropy of the family is monotone increasing in c.

Real dynamics




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

Real dynamics


Real dynamics


...requires to understandgeneral exponents

Real dynamics


Is the entropy of the family monotone increasing?


Real dynamics




Not rational maps of the sphere.

To study the dynamics in general and prove new rigidity results -

Real dynamics






Does the flow on expand its Teichmueller metric?

Real dynamics






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!

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