Functions Theory in Higher dimensions

Functions Theory in Higher dimensions Sirkka-Liisa Eriksson Tampere University of Technology Department of Mathematics P.O.Box 553 FI-33101 Tampere, Finland sirkka-liisa.eriksson@tut.… TUT

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Collaborator

Heinz Leutwiler University of Erlangen-Nürnberg Bismarckstrasse 1 12 D-91054 Erlangen, Germany E-mail:[email protected]

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Outline

1

Complex Plane

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Outline

1 2

Complex Plane Key Ideas in Higher Dimensions

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Outline

1 2 3

Complex Plane Key Ideas in Higher Dimensions Quaternions

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Outline

1 2 3 4

Complex Plane Key Ideas in Higher Dimensions Quaternions Cli¤ord algebra

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Outline

1 2 3 4 5

Complex Plane Key Ideas in Higher Dimensions Quaternions Cli¤ord algebra Hypermonogenic functions

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Outline

1 2 3 4 5 6

Complex Plane Key Ideas in Higher Dimensions Quaternions Cli¤ord algebra Hypermonogenic functions Integral formulas

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Outline

1 2 3 4 5 6 7

Complex Plane Key Ideas in Higher Dimensions Quaternions Cli¤ord algebra Hypermonogenic functions Integral formulas Riemannian manifolds

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Hyperbolic metric The hyperbolic surface measure and the volume measure are dσh =

dx dσ , dxh = n +1 . n xn xn

and the hyperbolic normal derivative is ∂ ∂ = xn . ∂nh ∂n

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Hyperbolic metric The hyperbolic surface measure and the volume measure are dσh =

dx dσ , dxh = n +1 . n xn xn

and the hyperbolic normal derivative is ∂ ∂ = xn . ∂nh ∂n geodesics are circular arcs perpendicular to the hyperplane xn = 0 (half-circles whose origin is on xn = 0 and straight vertical lines ending on the hyperplane xn = 0.

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The hyperbolic ball Bh (a, Rh ) with the hyperbolic center a and the radius Rh is an euclidean ball with the euclidean center ae = a0 + a1 e1 + ... + an

1 en 1

+ en an cosh Rh

and the euclidean radius Re = an sinh Rh .

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The hyperbolic ball Bh (a, Rh ) with the hyperbolic center a and the radius Rh is an euclidean ball with the euclidean center ae = a0 + a1 e1 + ... + an

1 en 1

+ en an cosh Rh

and the euclidean radius Re = an sinh Rh . the hyperbolic distance dh (x, a) between the points x = x0 + e1 x2 + ... + xn en and a = a0 + e1 a1 + ... + an en is Rh = dh (x, a) = arcosh δ(x, a), δ(x, a) =

S.-L. Eriksson (TUT)

jx

aj2 + 2an xn . 2xn an

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We map x and y to the points en and λen . Then dh (x, y ) = dh (en , λen ) =

Z λ dt 1

t

= ln λ,

1 1 δ(x, y ) = δ (en , λen ) = (λ + ) 2 λ 1 ln λ cosh dh (x, y ) = cosh ln λ = (e + e ln λ ) = δ(x, y ). 2

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We map the upperhalf space to the unit ball by T

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1

(z ) = (z

en )( en z + 1)

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1

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We map the upperhalf space to the unit ball by T

1

(z ) = (z

en )( en z + 1)

1

The unit ball to the upper half space by T (z ) = (z + en )(en z + 1)

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1

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We map the upperhalf space to the unit ball by T

1

(z ) = (z

en )( en z + 1)

1

The unit ball to the upper half space by T (z ) = (z + en )(en z + 1)

1

The Möbius transformation ϕ mapping x to en ! n ∑in=01 (zi xi ) ei + zn en . ϕ ∑ zi ei = xn i =0

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e = T 1 ( ϕ (y )) 2 B. Since Q e 2 B there is a rotation R Set Q e ) = µen for some µ 2 [0, 1]. such that R (Q Put ψ = T R T 1 ϕ. Then ψ is a Möbius transformation (as product of Möbius transformations) with ψ (x ) = T ψ (y ) = T

R R

T T |

1 ( ϕ (x ))

= en

1 ( ϕ (y ))

{z e Q

}

e ) ) = λen . = T (R ( Q |{z} µen

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e = T 1 ( ϕ (y )) 2 B. Since Q e 2 B there is a rotation R Set Q e ) = µen for some µ 2 [0, 1]. such that R (Q Put ψ = T R T 1 ϕ. Then ψ is a Möbius transformation (as product of Möbius transformations) with ψ (x ) = T ψ (y ) = T

R R

T T |

1 ( ϕ (x ))

= en

1 ( ϕ (y ))

{z e Q

}

e ) ) = λen . = T (R ( Q |{z} µen

ky x k2 = ky xˆ k2 1 1 2 (λ + λ )

T

1

(λen ) =

( λ 1 )2 ( λ +1 )2

=)

jy x j2 +2yn xn 2xn yn

=

= cosh dh (x, y )

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Hyperbolic function theory was initiated by Heinz Leutwiler around 1990.

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Hyperbolic function theory was initiated by Heinz Leutwiler around 1990. For any m 2 N, the power function f (x ) = x m is paravector-valued for any paravector x = x0 + x1 e1 + ... + xn en .

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In Higher Dimensions For any m 2 N, the power function x m satis…es the property xm = where h (x ) =

S.-L. Eriksson (TUT)

∂h ∂x0

n

∂h

∑ ei ∂xi

(1)

i =1

1 Re x m +1 . m+1

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In Higher Dimensions For any m 2 N, the power function x m satis…es the property xm =

∂h ∂x0

n

∂h

∑ ei ∂xi

(1)

i =1

where

1 Re x m +1 . m+1 h satis…es the preceding hyperbolic equation h (x ) =

xn 4 f

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(n

Hyperbolic

1)

∂f = 0. ∂xn

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H-solutions

Heinz Leutwiler initiated in 1992 the research of the functions f , called H-solutions, admitting locally the preceding property for some function h satisfying the equality

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Another characterization of H-solutions is that they are paravector-valued solutions f = u0 + u1 e1 + ... + un en of the following generalized Cauchy-Riemann equation ∂u ∂u0 ∑ni=1 ∂xii + (n 1) un ∂x0 ∂uk ∂ui i, k = 1, ..., n, ∂xk = ∂xi , ∂uk ∂u0 k = 1, ..., n. ∂xk = ∂x0 ,

xn

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= 0, (H)

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General assumptions

Let Ω be an open subset of Rn +1 .

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General assumptions

Let Ω be an open subset of Rn +1 . We consider function mapping f : Ω ! C `0,n , whose components are continuously di¤erentiable.

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Dirac operators

The left Dirac operators in C `0,n is de…ned by n

∂f Dl f = ∑ ei , ∂xi i =0

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n

Dl f =

∂f

∑ ei ∂xi ,

i =0

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Dirac operators

The left Dirac operators in C `0,n is de…ned by n

∂f Dl f = ∑ ei , ∂xi i =0

n

Dl f =

∂f

∑ ei ∂xi ,

i =0

The right Dirac operators by n

∂f Dr f = ∑ ei , ∂xi i =0

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n

Dr f =

∂f

∑ ∂xi ei .

i =0

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Theorem (Cauchy formula) Let Ω 2 Rn +1 be an open, K Ω a compact set with smooth boundary δK and ν its outer unit normal …eld. Then for any monogenic function f : Ω ! Cl0,n , that is Dl f = 0, we have f (q ) =

1 ω n +1

Z

δK

(p jp

q)

1

q jn

1

ν (p ) f (p ) dS,

for all q 2 K, where ω n +1 is the surface measure of the unit ball in Rn + 1 .

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Modi…ed Dirac Operators

Let Ω be an open subset of Rn +1 nfxn = 0g.

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Modi…ed Dirac Operators

Let Ω be an open subset of Rn +1 nfxn = 0g. l

r

The modi…ed Dirac operators Mkl , Mkr , M k and M k are introduced by 0

Mkl f (x ) = Dl f (x ) + k Qxnf , l

M k f (x ) = D l f (x )

0

Mkr f (x ) = Dr f (x ) + k Qf xn r

k Qxnf , Mk f (x ) = D r f (x )

k Qf xn .

where f 2 C 1 (Ω, C `0,n ) .

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Hypermonogenic functions De…nition Let Ω Rn +1 be open. A function f : Ω ! C `0,n is left hypermonogenic function, if f 2 C 1 (Ω) and Mkl f (x ) = 0 for any x 2 Ωn fxn = 0g . The right k-hypermonogenic functions are de…ned similarly. If k = n 1 left hypermonogenic functions are called hypermonogenic functions. Hypermonogenic functions were introduced by H. Leutwiler and S.-L. E in 2000.

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Basic properties

Hypermonogenic functions have values in the total Cli¤ord algebra C `0,n .

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Basic properties

Hypermonogenic functions have values in the total Cli¤ord algebra C `0,n . Paravector-valued hypermonogenic functions are H-solutions

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Basic properties

Hypermonogenic functions have values in the total Cli¤ord algebra C `0,n . Paravector-valued hypermonogenic functions are H-solutions A function f : Ω ! C `0,n is left k hypermonogenic function if and only if f is right hypermonogenic

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Basic properties

Hypermonogenic functions have values in the total Cli¤ord algebra C `0,n . Paravector-valued hypermonogenic functions are H-solutions A function f : Ω ! C `0,n is left k hypermonogenic function if and only if f is right hypermonogenic If f : Ω ! C `0,n is k hypermonogenic, then b f (b x ) is n + 1 b k-hypermonogenic in Ω = x 2 R jb x2Ω .

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Notably studied by

The theory is notably studied by H. Leutwiler J. Cnops P. Cerejeiras Th. Hemp‡ing P. Zeilinger Pernas, L. Junxia Li

S.-L. Eriksson (TUT)

S.-L. Eriksson Kettunen, J. Hirvonen, J. H. Orelma Ryan, J. Qiao, Yuying

S. Krausshar E. Lehman I. Ramadano¤ Bernstein, Sw. Laville, G. Xiaoli Bian

Hyperbolic

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Generalized Cauchy-Riemann equations

Theorem Let Ω be an open subset of Rn +1 and f : Ω ! C `0,n be a mapping 0 with continuous partial derivatives. The equation Df + k Qxnf = 0 is equivalent with the following system of equations ∂ (Q 0 f ) ∂xn ∂P 0 (f ) Dn 1 (Qf ) + ∂xn

Dn

S.-L. Eriksson (TUT)

1

(Pf )

Hyperbolic

0

+ k Qxnf = 0, = 0.

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Hyperbolic harmonic functions Theorem Let f : Ω ! C `0,n be twice continuously di¤erentiable. Then P Mk M k f = 4Pf Q Mk M k f = 4Qf

k ∂Pf xn ∂xn k ∂Qf Qf +k 2 xn ∂xn xn

If f is k-hypermonogenic, then k ∂Pf xn ∂xn k ∂Qf Qf +k 2 xn ∂xn xn

4Pf 4Qf S.-L. Eriksson (TUT)

Hyperbolic

= 0 = 0. 40 / 63

Laplace Beltrami equations

These are Laplace-Beltrami equations with respect to the Riemannian metric ∑n dx 2 ds 2 = i =02k i . xnn 1

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k-hyperbolic harmonic functions Mn

1

(fx ) = (Mn

S.-L. Eriksson (TUT)

1f ) x

for any paravector valued function f .

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k-hyperbolic harmonic functions Mn 1 (fx ) = (Mn 1 f ) x for any paravector valued function f . A twice continuously di¤erentiable function f : Ω ! Cl0,n is called k-hyperbolic harmonic if Mk M k f = 0.

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k-hyperbolic harmonic functions Mn 1 (fx ) = (Mn 1 f ) x for any paravector valued function f . A twice continuously di¤erentiable function f : Ω ! Cl0,n is called k-hyperbolic harmonic if Mk M k f = 0. Let f : Ω ! Cl0,n be twice continuously di¤erentiable. Then f is k-hypermonogenic if and only if f and xf are k-hyperbolic harmonic functions.

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k-hyperbolic harmonic functions Mn 1 (fx ) = (Mn 1 f ) x for any paravector valued function f . A twice continuously di¤erentiable function f : Ω ! Cl0,n is called k-hyperbolic harmonic if Mk M k f = 0. Let f : Ω ! Cl0,n be twice continuously di¤erentiable. Then f is k-hypermonogenic if and only if f and xf are k-hyperbolic harmonic functions. Let Ω be an open subset of Rn +1 and f : Ω ! C `0,n be twice continuously di¤erentiable. Then f is k-hypermonogenic if and only if there exists locally a k-hyperbolic harmonic mapping H with values in C `0,n 1 satisfying DH = f .

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Theorem Let U 1

Rn++1 be open. The following properties are equivalent:

h is hyperbolic harmonic on U.

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Theorem Let U 1 2

Rn++1 be open. The following properties are equivalent:

h is hyperbolic harmonic on U. h 2 C 2 (U ) and 1 h (a ) = σn sinhn Rh .

Z

∂Bh (a,Rh )

for all hyperbolic balls satisfying Bh (a, Rh )

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Hyperbolic

hdσh U.

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Theorem Let U 1 2

Rn++1 be open. The following properties are equivalent:

h is hyperbolic harmonic on U. h 2 C 2 (U ) and 1 h (a ) = σn sinhn Rh .

3

Z

∂Bh (a,Rh )

hdσh

for all hyperbolic balls satisfying Bh (a, Rh ) h 2 C 2 (U ) and 1 h (a ) = Vh (Bh (a, Rh ))

Z

Bh (a,Rh )

for all hyperbolic balls satisfying Bh (a, Rh ) RR Vh (Bh (a, Rh )) = σ 0 h sinhn tdt.

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Hyperbolic

U.

fdxh , U, where

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Examples

x m , m 2 Z, is hypermonogenic

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Examples

x m , m 2 Z, is hypermonogenic e x = ∑ k1! x k ,

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Examples

x m , m 2 Z, is hypermonogenic e x = ∑ k1! x k , 1 sin x = ∑ (2k + ( 1)k x 2k +1 , 1)!

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Examples

x m , m 2 Z, is hypermonogenic e x = ∑ k1! x k , 1 sin x = ∑ (2k + ( 1)k x 2k +1 , 1)!

cos x = ∑ (2k1 )! ( 1)k x 2k

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Examples

x m , m 2 Z, is hypermonogenic e x = ∑ k1! x k , 1 sin x = ∑ (2k + ( 1)k x 2k +1 , 1)!

cos x = ∑ (2k1 )! ( 1)k x 2k If f (z ) = ∑ ak z k is holomorphic and ak 2 R, then f (x ) = ∑ ak x k is hypermonogenic.

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Fueter Construction

If f = u + iv is holomorphic in an open set Ω e f (x ) = u x0 ,

q

C, then

x12 + ... + xn2 +

x1 e1 + .. + xn en q v x12 + .. + xn2

x0 ,

q

x12 + ... + xn2

is hypermonogenic.

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The k-hypermonogenic functions in an open subset Ω of Rn +1 form a right C `0,n 1 -module.

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The k-hypermonogenic functions in an open subset Ω of Rn +1 form a right C `0,n 1 -module. If f is a k-hypermonogenic function, then the function fen is hypermonogenic if and only if f = 0.

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The k-hypermonogenic functions in an open subset Ω of Rn +1 form a right C `0,n 1 -module. If f is a k-hypermonogenic function, then the function fen is hypermonogenic if and only if f = 0. A function f : Ω ! C `0,n is k-hypermonogenic if and only if the n is k-hypermonogenic. function fe xk n

S.-L. Eriksson (TUT)

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The k-hypermonogenic functions in an open subset Ω of Rn +1 form a right C `0,n 1 -module. If f is a k-hypermonogenic function, then the function fen is hypermonogenic if and only if f = 0. A function f : Ω ! C `0,n is k-hypermonogenic if and only if the n is k-hypermonogenic. function fe xk n

A function f : Ω ! C `0,n is left k-hypermonogenic if and only if the function fen satis…es the equation Dl g

S.-L. Eriksson (TUT)

ken Pg = 0. xn

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Stokes theorem

Theorem For Ω an open subset of Rn++1 (or Rn +1 ), K Ω a smoothly bounded compact set with outer unit normal …eld ν, and f , g 2 C 1 (Ω, C `0,n ), Z

∂K

Z

dσ P (g νf ) k = xn Q (g νf ) dσ =

∂K

S.-L. Eriksson (TUT)

Z

Z

K K

P (Mkr g ) f + gMkl f

dx xnk

Q (M r k g ) f + gMkl f dx.

Hyperbolic

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Theorem Let Ω be an open subset of Rn++1 and K Ω a smoothly bounded compact set with outer unit normal …eld ν. If f is hypermonogenic in Ω and y 2 K, then f (y ) =

2n 1 ω n +1

Z

K1 (x, y ) ν (x ) f (x )

K2 (x, y ) ν[ (x )f[ (x ) dσ.

∂K

where K1 (x, y ) = K2 (x, y ) =

S.-L. Eriksson (TUT)

ynn

jx jx

1

(x

y jn

1

ynn

1

y jn Hyperbolic

x (b 1

y)

jx jx

y)

1

ybjn

1

1

ybjn

1

.

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Theorem Let Ω be an open subset of Rn++1 and K Ω a smoothly bounded compact set with outer unit normal …eld ν. If f is hypermonogenic in Ω and y 2 K f (x ) =

2n 1 ω n +1

Z

k (x, y ) Q y ν0 (y )f 0 (y ) + xQ 0 (ν(y )f (y )) ∂K

where k (x, y ) =

=

1 22n

2y n n

xnn 1 yn

D

x

Z 1

(x jx

1

jy x j jx yb j

y)

y jn

1

s2 sn

n 1

ds

(x yb) 1 1 jx ybjn 1

are hypermonogenic with respect to x in Rn +1 n fy, ybg. S.-L. Eriksson (TUT)

Hyperbolic

!

!

.

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Theorem Let Ω be an open subset of Rn++1 and K Ω a smoothly bounded compact set with outer unit normal …eld ν. If f is continuous on Ω then 2n 1 g (x ) = ω n +1

Z

k (x, y ) Q y ν0 (y )f 0 (y ) + xQ 0 (ν(y )f (y )) dσ ∂K

is hypermonogenic in Rn++1 n∂K .

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Riemannian Manifolds Riemannian metric on an open set Ω ds 2 =

Rn has the form

n

∑

gij dxi dxj ,

i ,j =0

where (gij ) is a second order covariant tensor

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Riemannian Manifolds Riemannian metric on an open set Ω ds 2 =

Rn has the form

n

∑

gij dxi dxj ,

i ,j =0

where (gij ) is a second order covariant tensor We consider the conformal metric ds 2 =

2

∑ gii dxi2 ,

i =0

where gij = ρ2 (x ) δij . The mapping h : Rn ! Rn is conformal when h 0 (x ) 2 O (n ) jh0 (x )j

where O (n) is the set orthogonal transformations in Rn . S.-L. Eriksson (TUT)

Hyperbolic

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Covariant tensor

if x = (x0 , x1 , .., xn ) are the old coordinates and x = (x 0 , x 1 , .., x n ) the new coordinates in terms of the change then ! ∂xh ∂xk (g ij ) = ∑ ghk . ∂x i ∂x j h,k

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Dirac operator De…ne at each point x a tangent plane and extend it to a Cli¤ord algebra with the inner product (gij )

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Dirac operator De…ne at each point x a tangent plane and extend it to a Cli¤ord algebra with the inner product (gij ) If e1 , . . . , en +1 de…nes - at each point of the manifold - an orthonormal basis for the tangent space, the Dirac operator is de…ned by DlM f DrM f

n +1

=

∑ ei Oei f ,

i =1 n +1

=

∑ (Oei f )ei ,

i =1

where f has values in C `0,n +1 and Oei denotes the covariant derivative with respect to the vector …eld ei S.-L. Eriksson (TUT)

Hyperbolic

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Choosing the hyperbolic metric on the upper half space Rn++1 , de…ned by ∑n +1 dx 2 ds 2 = i =21 i , xn +1 an orthonormal basis for the tangent space is given by the tangent vectors ei = xn +1

S.-L. Eriksson (TUT)

∂ i = 1, . . . , n + 1. ∂xi

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Choosing the hyperbolic metric on the upper half space Rn++1 , de…ned by ∑n +1 dx 2 ds 2 = i =21 i , xn +1 an orthonormal basis for the tangent space is given by the tangent vectors ei = xn +1

∂ i = 1, . . . , n + 1. ∂xi

The Christo¤el symbols associated with the metric are Γkij =

1 xn +1

(δj,k δi ,n +1 + δk,i δj,n +1

δi ,j δk,n +1 ) ,

δi ,k denoting the Kronecker delta. S.-L. Eriksson (TUT)

Hyperbolic

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Oei ej = Oxn +1

∂ ∂xi

ej = xn +1 O

∂ ∂xi

ej

= xn +1 δi ,n +1

n +1 ∂ ∂ + xn2+1 ∑ Γkij ∂xj ∂xk k =1

= δi ,j en +1

δj,n +1 ei ,

for i, j = 1, . . . , n + 1.

S.-L. Eriksson (TUT)

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Using the product rule, ci ν1

νk

= Oe i ( e ν 1

eν ) 8k jνj > /ν < ( 1) ei eν en +1 if i 2 ν and n + 1 2 ν j j = Oe i ( e ν ) = if i 2 / ν and n + 1 2 ν ( 1) ei eν > : 0 otherwise n

f =

∑ f ν eν ν

and observing that

Oe i ( f ν e ν ) = Oe i ( f ν ) e ν + f ν Oe i e ν ∂fν = xn +1 eν + fν ci ν ∂xi S.-L. Eriksson (TUT)

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we conclude that n +1

Dhyp f

=

∑ ei Oei f

=

i =1

n +1

= xn +1

∂f

∑ ei ∂xi

i =1

∑

n +1 2 ν

S.-L. Eriksson (TUT)

(n + 1

∑

n +1/ 2ν

( 1)jνj jνj fν eν en +1

jνj) ( 1)jνj fν eν

Hyperbolic

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In case n = 2 we thus have for f

= f1 e1 + f2 e2 + f3 e3 +f12 e1 e2 + f13 e1 e3 + f23 e2 e3 +f123 e1 e2 e3

the system Dhyp f

S.-L. Eriksson (TUT)

= x3 Deucl f + 2f3 f13 e1 f23 e2 +f1 e1 e3 + f2 e2 e3 2f12 e1 e2 e3

Hyperbolic

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Set f = x3 g

S.-L. Eriksson (TUT)

Hyperbolic

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Split g : Ω ! C `n +1 , Ω open in Rn +1 , into its even and odd parts 1 1 geven = (g + g 0 ) , godd = (g g 0 ) 2 2

S.-L. Eriksson (TUT)

Hyperbolic

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Split g : Ω ! C `n +1 , Ω open in Rn +1 , into its even and odd parts 1 1 geven = (g + g 0 ) , godd = (g g 0 ) 2 2 Then x3 1 Dhyp f

even

= x3 Deucl godd + g3

x3 1 Dhyp f

odd

= x3 Deucl geven

S.-L. Eriksson (TUT)

Hyperbolic

g123 e12

g12 e1 e2 e3

59 / 63

Split g : Ω ! C `n +1 , Ω open in Rn +1 , into its even and odd parts 1 1 geven = (g + g 0 ) , godd = (g g 0 ) 2 2 Then x3 1 Dhyp f

even

= x3 Deucl godd + g3

x3 1 Dhyp f

odd

= x3 Deucl geven

g123 e12

g12 e1 e2 e3

The function godd satis…es the equation x3 Deucl w + Q w = 0

S.-L. Eriksson (TUT)

Hyperbolic

59 / 63

Split g : Ω ! C `n +1 , Ω open in Rn +1 , into its even and odd parts 1 1 geven = (g + g 0 ) , godd = (g g 0 ) 2 2 Then x3 1 Dhyp f

even

= x3 Deucl godd + g3

x3 1 Dhyp f

odd

= x3 Deucl geven

g123 e12

g12 e1 e2 e3

The function godd satis…es the equation x3 Deucl w + Q w = 0 The function geven satis…es the equation x3 Deucl w

S.-L. Eriksson (TUT)

P we3 = 0.

Hyperbolic

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References

Cnops, J., Hurwitz pairs and applications of Möbius transformation, Habilitation thesis, Univ. Gent, 1994. Eriksson-Bique, S.-L, On modi…ed Cli¤ord analysis, Complex Variables,Vol. 45, (2001), 11–32. Eriksson-Bique, S.-L., k hypermonogenic functions. In Progress in Analysis, Vol I, World Scienti…c (2003), 337–348.

S.-L. Eriksson (TUT)

Hyperbolic

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Eriksson, S.-L., Integral formulas for hypermonogenic functions, Bull. Bel. Math. Soc. 11 (2004), 705–717. Eriksson-Bique, S.-L. and Leutwiler, H. , On modi…ed quaternionic analysis in R3 , Arch. Math. 70 (1998), 228–234. Eriksson-Bique, S.-L. and Leutwiler, H., Hypermonogenic functions. In Cli¤ord Algebras and their Applications in Mathematical Physics, Vol. 2, Birkhäuser, Boston, 2000, 287–302.

S.-L. Eriksson (TUT)

Hyperbolic

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Eriksson-Bique, S.-L. and Leutwiler, H., Hypermonogenic functions and Möbius transformations, Advances in Applied Cli¤ord algebras, Vol 11 (S2), December (2001), 67–76. Eriksson, S.-L. and Leutwiler, H., Hypermonogenic functions and their Cauchy-type theorems. In Trend in Mathematics: Advances in Analysis and Geometry , Birkhäuser, Basel/Switzerland, 2004, 97–112. Eriksson, S.-L. and Leutwiler, H., Contributions to the theory of hypermonogenic functions, Complex Variables and elliptic equations 51, Nos. 5-6 (2006), 547–561.

S.-L. Eriksson (TUT)

Hyperbolic

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Leutwiler, H., Modi…ed Cli¤ord analysis, Complex Variables 17 (1992), 153–171. Leutwiler, H., Modi…ed quaternionic analysis in R3 , Complex Variables 20 (1992), 19–51. Eriksson, S-L., Bernstein, S.,Ryan, J., Qiao, Y., Function Theory for Laplace and Dirac-Hodge Operators in Hyperbolic Space, Journal d’Analyse Mathematique, Vol 98 (2006) 43–64.

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Hyperbolic

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