International Conference on Algebras and Lattices

Prague, Czech RepublicJune 21-25, 2010

Algebra of the solutions to the Beltrami equation

Algebra of the solutions to the Beltrami equation

Eduard YakubovH.I.T.- Holon Institute of Technology

Israel

Jointly with U.Srebro (Technion) and

D.Goldstein (H.I.T.)

Prague-2010

The authors

would like to thank

the organizers

for invitation

and

hospitality!

Notation:

)(.3

),(.2

},,|{.11

zfivuw

Cf

yxiyxz

CC

RC

Complex derivatives:

)(2

1:

)(2

1:

:

:

yxz

yxz

yyy

xxx

ifff

ifff

ivuf

ivuf

Complex dilatation:

0,0

0,)(

)(

:)(

z

zz

z

f

f

fzf

zf

z

Jacobian of

)|)(|1(||

||||)(22

22

zf

ffzJ

fz

zzf

:f

Sense-preserving mappings

Sense-reversing mappings

1||0.. ffJpsf

1||0.. ffJrsf

The Beltrami Equation

.

)(,0)(

.)(

)()()()(

EquationRiemannCauchy

thetoreduceszFor

giveniszHere

zfzzf zz

B

B

Historical RemarksGauss (1880) Isothermal Coordinates - Real Analytic case

Korn (1914)&Lichtenstein (1916) - Holder Cont’ case

Lavrent’ev (1935) - Continuous case

Morrey (1938) - Measurable case

Ahlfors, Bojarski,Vekua (mid 50’s) - Measurable caseSingular Integral Methods

Among contributors to (B) theory

Ahlfors, Andrean-Cazacu, Astala, Belinski, Bers, Bojarski, Brakalova, Danilyuk, David, Dzuraev, Earle, Gutlyanskii, Heinonen, Iwaniec, Jenkins, Krushkal, Kuhnau, Lavrent’ev, Lehto, Martin, Martio, Migliaccio, Miklyukov, Moscariello, Morrey, Mueller, Painvarinta, Pesin, Pfluger, Reich, Ricciardi, Ryazanov, Salimov, Sastry, Shabat, Sheretov, Srebro, Strebel, Sugava, Sullivan, Suvorov, Sverak, Teichmuller, Tukia,Vekua, Virtanen, Volkoviski, Vuorinnen, Walczak, Zhong Li, Yakubov, Zahirov, Zorich

Applications

Uniformization

Robotics Elasticity Control Theory String Theory Hydro - dynamics Kleinian Groups Riemann Surfaces Teichmuller Spaces Holomorphic Motions Complex Dynamics Differential Geometry Conformal Field Theory Quasi-conformal mappings Low Dimensional Topology

Classification of (B)

The Classical Case:

The Relaxed Classical Case:

The Alternating Case:

1||||

1||||.,.1|| ea

C

C

ofpartotherinea

and

ofpartainea

.1||

..1||

Solutions in the measurable case

A continuous mapping is called

a solution to (B) if and satisfy (B) a.e.

ACL mean absolutely continuous on a.e. horizontal and vertical lines

ACLfCC:f

zz ff ,

Open and discrete solutions of (B)

A solution is called open if is open whenever is open.

A solution is called discrete if is a discrete set for every .

f )(Ef

E

f )(1 wf

Cw

Main Problems

Existence

Uniqueness

Regularity

Representation of the solution set

Removability of isolated singularities

Boundary behavior

Mapping properties

Existence and uniqueness of solutions

THEOREM

Let be a measurable function with

Then:

1. (B) has a homeomorphic solution .

2. is unique up to a post-composition by a conformal map

3. generates the set of all open, discrete and s.p. solutions, i.e. for every open, discrete and s.p. solution there is an analytic function such that

f

f

f

1||||

gh fhg

Algebra of solutions

THEOREM 1

For every measurable function the set

of the solutions of equation (B) is an algebra

(with respect to point-wise addition & multiplication)

Proof: Straightforward calculation

)(S

Algebra of solutions

THEOREM 2

Let be a measurable function with .

The set of open, discrete and s.p. solutions

of equation (B) is a sub-algebra of .

Proof: Based on Stoilow’s decomposition theorem

)(0 S)(z 1||||

)(S

Algebra of solutions

THEOREM 3

Let be measurable functions.

If a.e. then .

)(),( zz C )()( SS

Algebra of solutions

THEOREM 4Let be a measurable function with .Then the algebra of open, discrete and s.p. solutions of (B) is isomorphic to the algebra of analytic functions in .

Proof: Based on Stoilow’s decomposition theorem

1|||| )(0 S

C

Algebra of solutions

Remark: Similar results are also true in

the alternating case for (B) which has :

1. Proper folding solutions

2. Proper (p,q)- cusp solutions

0Im,

0Im,)()( 0 zz

zzzfzf

)12,(

0Im,||

0Im,||

)()(

1

1

,

Nqp

zz

z

zz

z

zfzf

q

q

p

p

qp

Algebra of solutions

3. Proper straight umbrella solutions

mappinginjectivewisepiece

continuousaisSSHere

z

zzzfzf

11

0

:

)||

(||)()(

Uniqueness and Representation Properties in the alternating case

The following solutions are unique and generates the solution set:• Proper folding solutions• Proper (p,q)-cusp solution with |p-q|=2 • Proper umbrella solution of degree 1 or -1

Corollary: (B) cannot have both folding solutions and (3,1)-cusp solutions

Algebra of solutions

Open problems:1. Characterize algebras of solutions in the relaxed

classical case:

2. What are connections between topological properties of solutions and corresponding algebras?2. Describe ideals of algebras of solutions according to . 3. Describe lattices of the solutions according to .

1||||,..1|| ea

Reference:

1. U.Srebro and E. Yakubov, Beltrami Equations, Handbook in Complex Analysis, 2005, Elsevier2. T.Iwaniec and G.Martin, The Beltrami Equation, 2008, Memories of AMS3. O.Martio, V.Ryazanov, U.Srebro and E.Yakubov, Moduli in Modern Mapping Theory, 2009, Springer Monographs in Mathematics4. V. Gutlyanskii, V.Ryazanov, U.Srebro and E.Yakubov, The Beltrami Equation: A Geometric Approach, Springer Monographs in Mathematics (to appear)

Thank you

for your attention !