Energy-minimal di ff . between doubly conn. Riemann surfaces David - PowerPoint PPT Presentation

Energy-minimal di ff . between doubly conn. Riemann surfaces David Kalaj University of Montenegro Osaka, January, 2013 FILE: OsakaKalaj1.tex 2013-1-6, 12.28 David Kalaj Energy-minimal di ff . between doubly conn. Rieman 1/47

Harmonic mappings between Riemann surfaces Let ( M , � ) and ( N , ⇢ ) be Riemann surfaces with metrics � and ⇢ , respectively. If a mapping f : ( M , � ) ! ( N , ⇢ ) is C 2 , then f is said to be harmonic (to avoid the confusion we will sometimes say ⇢ -harmonic) if f zz + (log ⇢ 2 ) w � ff z f ¯ z = 0 , (1) where z and w are the local parameters on M and N respectively. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 2/47

Also f satisfies (1) if and only if its Hopf di ff erential Ψ = ⇢ 2 � ff z f ¯ z is a holomorphic quadratic di ff erential on M . For g : M 7! N the energy integral is defined by M ( | @ g | 2 + | ¯ R @ g | 2 ) dV � . E ⇢ [ g ] = Then f is harmonic if and only if f is a critical point of the corresponding functional where the homotopy class of f is the range of this functional. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 3/47

Mappings of finite distortion A homeomorphism w = f ( z ) between planar domains Ω and D has finite distortion if a) f lies in the Sobolev space W 1 , 1 loc ( Ω , D ) of functions whose first derivatives are locally integrable, and b) f satisfies the distortion inequality | f ¯ z |  µ ( z ) | f z | , 0  µ ( z ) < 1 almost everywhere in Ω . Such mappings are generalizations of quasiconformal homeomorphisms. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 4/47

The Nitsche conjecture The conjecture in question concerns the existence of a har. homeo. between circular annuli A ( r , 1) and A ( ⌧ , � ) , and is motivated in part by the existence problem for doub.-conn. min. surf. with prescribed boundary. In 1962 J. C. C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). David Kalaj Energy-minimal di ff . between doubly conn. Rieman 5/47

Indeed Nitsche observed that a radial harmonic mapping f : A ( r , 1) ! A ( ⌧ , � ) , i.e. satisfy. the condition f ( se it ) = f ( s ) e it ( f ( z ) = az + b / ¯ z ), is a homeomorphism if and only if � ⌧ � 1 � 1 � r + r (Nitsche bound). 2 Then he conjectured that for arbitrary harmonic homeomorphism between annuli we have Nitsche bound. Some partial solutions are presented by Weitsman, Lyzzaik, Kalaj (2001,2003). David Kalaj Energy-minimal di ff . between doubly conn. Rieman 6/47

The Nitsche was right The Nitsche conjecture for Euclidean harmonic mappings is settled recently by Iwaniec, Kovalev and Onninen (2010, JAMS), showing that, only radial harmonic mappings ⇣ ⌘ ⇣ � ! h ( ⇣ ) = C , C 2 C , ! 2 R , ⇣ | C | (1 � ! ) = � , which inspired the Nitsche conjecture, make the extremal distortion of rounded annuli. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 7/47

Radial ⇢ � harmonic mappings We state a similar conjecture (see Kalaj, arXiv:1005.5269) with respect to ⇢ � harmonic mappings. In order to do this, we find all examples of radial ⇢ -harmonic maps between annuli. We z = se it in harm. put w ( z ) = g ( s ) e it , eq. h zz + (log ⇢ 2 ) w � hh z h ¯ z = 0 where g is an increasing or a decreasing function. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 8/47

The resulting functions are w ( z ) = g ( s ) e it , z = se it , where g is the inverse of ✓R � ◆ dy p h ( � ) = exp , ⌧  �  � , � y 2 + c % 2 % = 1 / ⇢ . Moreover they are homeomorphisms i ff we have ⇢ � Nitsche bound: Z ⌧ ⇢ ( y ) dy r � exp( ) . p y 2 ⇢ 2 ( y ) � ⌧ 2 ⇢ 2 ( ⌧ ) � David Kalaj Energy-minimal di ff . between doubly conn. Rieman 9/47

Generalization of Nitsche conjecture Let ⇢ be a radial metric. If r < 1 , and there exists a ⇢ � harmonic mapping of the annulus A 0 = A ( r , 1) onto the annulus A = A ( ⌧ , � ) , then there hold the ⇢ � Nitsche bound. Notice that if ⇢ = 1 , then this conjecture coincides with standard Nitsche conjecture. For some partial solution see Kalaj (2011, Israel J. of Math) David Kalaj Energy-minimal di ff . between doubly conn. Rieman 10/47

Another justification of the previous conjecture Assume that a homeomorphism f : A (1 , r ) ! A ( ⌧ , � ) has a finite distortion and minimize the integral R Ω K ( z , f ) ⇢ 2 ( z ) dxdy . Here means K ⇢ [ f ] = K ( z , f ) = ( | f z | 2 + | f ¯ z | 2 ) / ( | f z | 2 � | f ¯ z | 2 ) is Distortion function. If the annuli A (1 , r ) and A ( ⌧ , � ) are conformaly equivalent then the absolute minimum is achieved by a conformal mapping. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 11/47

If the annuli A (1 , r ) and A ( ⌧ , � ) are not conformaly equivalent then the absolute minimum is achieved by a homeomorphism f whose inverse h is a ⇢ harmonic mapping between A ( ⌧ , � ) and A (1 , r ) if and only if the annuli satisfy the ⇢ � Nitsche bound. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 12/47

The previous result is generalization of a result of Astala, Iwaniec and Martin (2010, ARMA) and has been obtained by Kalaj (2010, arXiv:1005.5269). If the annuli A (1 , r ) and A ( ⌧ , � ) do not satisfies the ⇢ � Nitsche condition, then no such homeomorphisms of finite distortion exists between them. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 13/47

Energy minimizers, radial metric, (Kalaj, arXiv:1005.5269) Let ⇢ be a regular metric. Within the Nitsche rang, for the annuli A and A 0 , the absolute minimum of the energy h 2 W 1 , 2 ( A , A 0 ) is integral h ! E ⇢ [ h ] , attained by a ⇢ � Nitsche map h c ( z ) = q � 1 ( s ) e i ( t + � ) , z = se it , � 2 [0 , 2 ⇡ ) , where ✓R s ◆ dy p q ( s ) = exp , ⌧ < s < � . � y 2 + c % 2 David Kalaj Energy-minimal di ff . between doubly conn. Rieman 14/47

Energy minimizers, Euclidean metric Let A and A 0 be doub. conn. domains in compl. plane such that Mod ( A )  Mod ( A 0 ) . The absolute minimum of the energy integral h 2 W 1 , 2 ( A , A 0 ) is attained by h ! E [ h ] , an Euclidean harmonic homeomorphism between A and A 0 . (Iwaniec, Kovalev, Onninen, Inventiones, (2011)) David Kalaj Energy-minimal di ff . between doubly conn. Rieman 15/47

Energy minimizers, Arbitrary metric with bounded Gauss curvature Let A and A 0 be doub. conn. domains in Riemann surfaces ( M , � ) and ( N , ⇢ ) , such that Mod ( A )  Mod ( A 0 ) . The absolute minimum of the energy h 2 W 1 , 2 ( A , A 0 ) is integral h ! E ⇢ [ h ] , attained by an ⇢ � harmonic homeomorphism between A and A 0 . (See Kalaj, arXiv:1108.0773) David Kalaj Energy-minimal di ff . between doubly conn. Rieman 16/47

The proof uses The so called deformations D ⇢ ( Ω , Ω ⇤ ) which make a compact family. A modification of Choquet-Rado-Kneser theorem for Riemann surfaces with bounded Gauss curvature The fact that energy integral is weak lower semicontinuous. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 17/47

And the key lemma Let Ω = A ( r , R ) be a circular annulus, 0 < r < R < 1 , and Ω ⇤ a doubly connected domain. If h 2 D ⇢ ( Ω , Ω ⇤ ) is a stationary deformation, then the Hopf di ff erential of the mapping h : ⇢ 2 ( h ( z )) h z h ¯ z ⌘ c z 2 in Ω where c 2 R is a constant. This lemma follows by a result of Jost. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 18/47

Conjecture a) If f is a minimizer of the energy E ⇢ between two doubly connected domains Ω , ( ⌧ = Mod ( Ω ) ) and Ω ⇤ , then f is harmonic and K ( ⌧ ) -quasiconformal if and only if ⌧ is smaller that the modulus ⌧ ⇧ of critical Nitsche domain A ( ⌧ ⇧ ) . David Kalaj Energy-minimal di ff . between doubly conn. Rieman 19/47

Conjecture b) b) Under condition of a) we conjecture that c ⇧ < 0 and r c ⇧ � c r c ⇧ K ( ⌧ ) = max { , c ⇧ � c } . (1) c ⇧ Notice that, this is true if the image domain is a circular annulus with a radial metric. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 20/47

Quasiconformal harmonic mappings (HQC) The first author who studied the class HQC is Olli Martio on 1968 (AASF). The class of HQC has been studied later by: Hengartner, Schober, Mateljevi´ c, Pavlovi´ c, Partyka, Sakan, Kalaj, Vuorinen, Manojlovi´ c, Nesi, Alessandrini, Boˇ zin, Markovi´ c, Wan, Onninen, Iwaniec, Kovalev etc. David Kalaj Energy-minimal di ff . between doubly conn. Rieman 21/47

Lipschitz and co-Lipschitz mappings Let w : Ω ! D be a mapping between two domains Ω and D . Then w is called Lipschitz (co-Lipschitz) continuous if there exists a constant C > 1 ( c > 0 ) such that | w ( z ) � w ( z 0 ) |  C | z � z 0 | , ( c | z � z 0 |  | w ( z ) � w ( z 0 ) | ) z , z 0 2 Ω . David Kalaj Energy-minimal di ff . between doubly conn. Rieman 21/47