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Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

2012-09-19T01:57:26Z

I asked the question before, but didn't get any reply, so I took the liberty to ask again.

Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is the following true ?

(Question 1) $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}$ for some $K>0$ independent of $z\in \mathbb{D}, \zeta \in S^1$.

Actually, by using weak maximum principle on $|z-\zeta|^{\alpha} - H(z,\zeta)$, where $H(z,\zeta)= \int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| $ is the complex harmonic extension of $t\to |t-\zeta|^{\alpha}$ at $z$, I am getting $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \geq |z-\zeta|^{\alpha}$, which is quite the contrary ! Any help or suggestions ? Thank you !

(Question 2)And also, as a seemingly related question can we say that if $f\in C^{0,\alpha}(S^1)$, then its complex harmonic extension $\int_{t\in S^1} f(t).p(z,t)|dt|$ is $C^{0,\alpha}(\mathbb{D}) ?$ Thanks.

On mentioning recommenders' names in cover letter for postdoctoral applications

2013-03-31T00:18:29Z

If I want to apply for a postdoctoral job, can I mention the name of my recommenders in my cover letter just to bolster my application, particularly when I am sure that the people who will read my application will know the recommenders personally? I know that they will actually read the recommendation letter themselves, but 1) I don't know how the selection process for postdocs actually works (particularly in Europe) and 2) I am afraid that one of my recommenders' letter might be a bit later than deadline. These are the reasons why I am thinking about it.

It will be great to have your opinion. Thanks.

Added by Theo JF: In your response, please clearly state your background and knowledge — have you served on postdoc hiring committees in US? in Europe? Certainly the advise that graduate students pass on to each other can be correct and valuable, but it's important to distinguish between "here's what I did and it seemed to work, and here's what my friend did" and "here's what I know from having reviewed three hundred applications, having spoken with twenty other committee members over the years, and having been involved with seven postdoc hires".

Softwares for drawing hyperbolic surfaces , closed, with boundaries or with punctures ?

2012-05-03T16:38:11Z

In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic surfaces, either closed, or with boundaries or punctures ? Are there any such freely downloadable softwares that I can use to draw such diagrams ? Please let me know if you know one. I apologize if this is off-topic. Thank you !

A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

2011-02-10T14:56:04Z

Hello,

I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, with continuity and uniqueness/injectivity would give the result after applying Brower's invariance of domain. I have a question about this proof and its notation.

First, to prove the continuity of $\omega$ , we need to have a topology on $ Teich(S_g) $. I know we can define it via Teichmuler metric, but they seems to have not defined it in this chapter ? So, how much structure about the set $ Teich(S_g) $ should I assume and exactly what topology are they using ?

They are actually trying to prove here that $ \kappa=exp(d_{Teich})$ [ still undefined so far ] is continuous. What did they mean by the symbol $ DF( \pi_1(S_g), PSL(2,R) ? $ I know the individual meanings of these notations, but I am not familiar with this symbol. What is $ F $ though ?

I also cannot follow " the marked Fundamental domains for these representations can be made K-quasiconformally equivalent for any $ K > 1$ by taking $Y'$ sufficiently close to $Y $ : what is a marked Fundamental domain, is it just any fundamental domain for the marked surface ? And how can they be made $ K $ q.c equivalent for any $ K> 1$ ?

Also, "By teichmullers uniqueness theorem, the infimum $\kappa(Y) $ is realized by some $K_h$ : why os it true for ALL $Y$ ? isn't it true for only the image of $\omega$ by Teichmuller uniqueness , because we are constructing a new surface from a given Riemann surface and a q.d. such that the identity map becomes $K q.c $ Teichmuller map ?

A somewhat detailed explanation would be appreciated, thanks.

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

2013-02-26T03:10:14Z

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following

Theorem: Fix $k \geq 0, 0<\alpha<1$. Let $f\in C^{k,\alpha}(S^1)$. Then its harmonic extension $H(f)$, which is the solution to the Dirichlet problem on the unit disk $D$ with boundary value $f$, is in $C^{k, \alpha}(D)$.

My main question is: is the above true for $n\geq 2$ as well? Any refernces/ suggestions?

While I don't know exactly a complete reference for the proof, but I have read the following theorem mentioned in the book "Boundary Behaviour of Conformal maps" by Christian Pommerenke which states:

Let $F:D\to\Omega $ be a conformal homeomorphism of $D$ onto a Jordan domain $\Omega$ whose boundary curve $\partial\Omega$ has a $C^{k,\alpha}$ -parametrization. Then $f\in C^{k, \alpha}(D)$. Note that any conformal homeomorphism $F$ of $D$ onto a Jordan domain extends to the boundary of $D$, by Caratheodory's extension theorem.

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

2012-11-12T17:51:30Z

My question is about the precise definition regarding the following:

Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a positive real number for every $b$ in $S^1$. Now I want to define when $f$ is $C^{1,\alpha}, 0<\alpha<1,$ near $1\in S^1$. I know one definition can be $f'(a)-f'(1)= O|a-1|^{\alpha}, i.e. |f'(a)-f'(1)|\le K.|a-1|^{\alpha}$. But I was wondering whether we can use the following alternate definition, using only information on $f$, but not on $f'$, motivated by $C^{1,\alpha}$-maps on $\mathbb{R}^1$:

1) Can we say $f$ is $C^{1,\alpha}$ near $1$ if $|f(a)-f(1)-f'(1)(a-1)|= O(|a-1|^{1+\alpha})$ ?

2) I have also another related question. let $F$ be a $C^1$ diffeomorphism on an open set containing the closed unit disk $\bar{D}$ and $F\in C^2({\mathbb{D}})$, and let $lim_{z\to1}F_z(z)=p, lim_{z\to1}F_{\bar{z}}(z)=q$. then can we say that $F$ is $C^{1,\alpha}$ near $1$(near in the sense of the topology of $\bar{D}$) if $|F(z)-F(1)-p(z-1)-q(\bar{z}-1)|=O|z-1|^{1+\alpha}$.

If the above are not correct, could you please give me or refer to me an alternate definition for each of the above ? Thank you.

Answer by Analysis Now for Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

2012-11-12T18:05:57Z

It is true, you have to use Kellog-Warschowski's theorem (wikipidea, will give link soon),for $C^{0,\alpha}$ -maps on $S^1$, and note that $|t-\zeta|^{\alpha}$ is a $C^{0,\alpha}(S^1)$ map, with the Holder constant being independent of $\zeta \in S^1$.

Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

2012-06-21T01:40:35Z

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:

Consider an orientation-preserving quasiconformal homeomorphism $f$ of the open unit disk $\mathbb{D}\subset \mathbb{C}$ with the complex dilatation/Beltrami cofficient $\mu, ||\mu||_{L^\infty(\mathbb{D})}<1, \mu \in C^0(\mathbb{\bar{D}})$ , i.e. $\mu$ is continuous on the closed unit disk $\mathbb{\bar{D}}$. Is it true that the restriction of $f$ on the boundary $S^1$ has continuous (ordinary) derivative on $S^1$, i.e., is $f\in C^1(S^1)$ ?

I guess one might start with continuously extending $\mu$ to all of $\mathbb{C}$, then solve the Beltrami equation on $\mathbb{C} $ with that extended $\mu$, but then I am getting stuck:because I guess the solution to this new equation might not be $C^1(\mathbb{C})$ ?? ( Look at Examples 15.1 in the book "Elliptic PDE and Quasiconformal Mappings" by K. Asltala, T. Iwaniec and G. Martin.

http://books.google.com/books?id=5aOgM9XRiXUC&printsec=frontcover&hl=sl#v=onepage&q&f=false

May be, to start with, we can ALSO assume that $\mu=0$on $S^1$. Then is the answer to my question yes ?

Could you please cite a reference of your proof to my question or give a counterexample ? Thanks a lot !!

Question on "publication List" for applying to post-doctoral jobs

2012-10-26T05:21:40Z

1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on arXiv is always good, because people might search for the arXiv -ed papers, but how much difference is publication on arXiv going to make ? What if I prepare a publication list (in .pdf) of accepted paper(s), and submitted paper(s) and paper(s) in preparation and send it to the employers ? Will that be sufficient or as good as putting them on arXiv ? (I will also send them a link to my site containing the publication list, including the downloadable links).

The situation is: I can put one of my papers in arXiv, but the others are either in collboration or deal with problems that stem from a question raised by collaborator(s) and the collaborator(s) did't agree to put on arXiv right away. Hence I am asking.

2) Also what exactly should I write in the "list of publications", the name of the paper, author(s) and its status(accepted/submitted/in preparation), that's all or or should I briefly also describe its content/abstract (to make it more informational) ?

Thank you.

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

2012-10-02T01:47:29Z

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :

what is the limit of $\frac{\int_{S^1}|t-a|^{1 + \alpha}.p(z,t)|dt|}{|z-a|}$ as $z \to a, z\in \mathbb{D}$. I am tending to believe that the limit is zero, because of the following reason :

Take any $0<\alpha' < \alpha$, then $t \mapsto |t-a|^{1 + \alpha'}\in C^{1,\alpha'}(S^1)$.Therfore by Kellog's (or by Kellog-Warschawski's) theorem, its harmonic extension,extended by $H(z)= \int_{S^1}|t-a|^{1 + \alpha'}.p(z,t)|dt|$on $\mathbb{D}$ is $C^{1,\alpha'}(\mathbb{D})$, therefore the harmonic extension is Holder continuous, that is :

$\frac{\int_{S^1}|t-a|^{1 + \alpha'}.p(z,t)|dt|}{|z-a|} \leq M \equiv M(\alpha')$ [Note that, at $a\in \partial \mathbb{D}, H(a)=0$].

But then there should be the 'effect' of this "extra" $\alpha - \alpha'$ in the integration, which, heuristically, should make the limit go to zero. But I am not sure how to prove that ? Is it right at least ? Any help will be highly appreciated, thank you !

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

2012-09-24T02:05:45Z

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $f$, which can also be shown to be a homeomorphism of $\bar{\mathbb{D}}$ [Choquet's theorem].

Now,it follows from a well-known theorem of Kellog-Warschowski that if $f\in C^{k,\alpha}(\mathbb{S^1})$, then $H(f) \in C^{k,\alpha}(\mathbb{D})$.

My question is : assuming $k=1$, what is the limiting matrix of the (total) derivative matrix $DH(f)_p$ as $p\to \zeta \in \mathbb{S^1},p \in \mathbb{D}$ ? From some other kind of extensions I have seen before, I would take a guess that $DH(f)_p \to f'(\zeta).Id$ as $p\to \zeta$, where $Id$ denotes the $2$ by $2$ identity matrix.

A reference to your answer will be highly appreciated $!$

Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

2012-08-29T19:47:58Z

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : A) what is the maximal subset of the real line $R$, where these two lifts are equal ? Also, conversely,B) if $\tilde{f}=\tilde{g} $on $R$, can we say that $f,g$ are homotopic ? What is the case for the underlying Fuchsian group $\Gamma$ for $X$ when these two lifts are equal on the whole real line $R$ ?

I have been trying to give the proof of A) of the proof of this myself using the concept of limit sets of the Fuchsian group $\Gamma$. To solve this, we could assume,after post-composing with $g$, that $f$ is homotopic to $Id$, without loss of generality. So we need to prove that $\tilde{f}=Id$ on $R$. The also, the converse question becomes: can we say that $f$ is homotopic to $Id$ if $\tilde{f}=Id$ on R ?

Effort of the proof of A):

Since $f$ is homotopic to $Id$, at the level of fundamental group with a fixed base point, $f_*=Id$ if $\tilde{f}$ is $Id$ on $R$.

But on the other hand, at the level of of deck transformation groups, $f_*$ is nothing but conjugation by $f$, am I right, (please correct me if I am not) ? So we have $\tilde{f}\circ \gamma \circ \tilde{f}^{-1}= \gamma \forall \gamma \in \Gamma$, which gives me that $\tilde{f}=Id$ only on the limit set $\Lambda(\Gamma)$ of $\Gamma$ . But when should it happen that $\tilde{f}=Id$ on $R$, if $\Lambda(\Gamma)\ne R$ ?

Also, my related question would be what are all the case of the Fuchsian group $\Gamma$ where the limit set $\Lambda(\Gamma)= R$ ?

$C^{1,\alpha}$-regularity of certain function related to harmonic extension appearing as inverse function

2012-08-11T18:09:08Z

The question I am going to ask refers to the following paper :

http://www.springerlink.com/content/hw88761257310165/.

For a fixed orientation-preserving homeomorphism $f$ of the unit circle $S^1$,define the function $G: \mathbb{D}\times \mathbb{D} \to \mathbb{D}$ by: $G(z,w):= \int_{S^1}\frac{f(t)-w}{1-\bar{w}f(t)}. p(z,t)|dt|$, where $p(z,t)= \frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2} $ is the Poisson kernel . It is shown in the above paper that, for a fixed $z\in S^1$, $G(z,w)$ has a unique $w$ -zero in the open unit disk $\mathbb{D}$, which we denote by $\Phi(f)(z)$.

My question is : if I assume that $f\in C^{1,\alpha}(S^1)$, then is $\Phi(f)\in C^{1,\alpha}(\mathbb{D})$ ? The reason I am even suspecting this to be true is the following : Note that for fixed $w\in \mathbb{D}, h:t\mapsto\frac{f(t)-w}{1-\bar{w}f(t)}$ is the left-composition of $f$ with a conformal automorphism $c: p\mapsto \frac{p-w}{1-\bar{w}.p}$of $\mathbb{D}$, hence $h$ is $C^{1,\alpha}(S^1)$, because $f\in C^{1,\alpha}(S^1)$ and $c\in C^{\infty}(\mathbb{\bar{D}})$ [ c has pole outside the unit disk]. Also, note that for fixed $w$, $G(z,w)$ is the complex harmonic extension of the circle homeomorphism $ h: t\mapsto\frac{f(t)-w}{1-\bar{w}f(t)}$. So, by Kellog's theorem (cited in any standard PDE book, see for example Gilberg-Trudinger), we have the complex harmonic extension $G$ of $h$ is $C^{1,\alpha}(\mathbb{D})$.

But the above automatically does $\textbf{not}$ guarantee that if $f\in C^{1,\alpha}(S^1)$, then $\Phi(f)\in C^{1,\alpha}(\mathbb{D}) $. Note that $\Phi(f)$ comes an implicit function of something we know $C^{1,\alpha}$-regularity about. Can we say anything about $\Phi(f)$ ? I understand the above comment is slightly vague, but I would appreciate if you have seen some similar situations like this before and can cite it or help me with this. Thank you !

Quick references/sources for the hyperbolic Riemann Surfaces with boundary

2012-07-23T03:39:22Z

Hello,

Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of hyperbolic Riemann surfaces with boundary look like (what kind of subsets of the Poincare disk are the fundamental domains?) and the hyperbolic structures on surfaces with boundary, the fundamental groups of them (are they Fuchsian groups?), the characterization of elements in the case (for example, characterization of elements of the fundamental group according to the geometry of the surface with boundary, e.g. parabolic elements correspond to cusp, hyperbolic elements correspond to simple closed non-trivial geodesics) etc.

A quick introduction (or reference) from which I can gather enough material within 1 or 2 weeks or so would be highly appreciated !

Thank you very much.

Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$

2012-07-08T23:41:58Z

Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have the concept of $\Gamma$-compatible Beltrami cofficients $\mu$ on $D$ which satisfy: $\mu(z)=\mu(\gamma(z)).\frac{\bar{\gamma'(z)}}{\gamma'(z)}$ (which corresponds to Beltrami differential forms on the Riemann surface $X=D/{\Gamma}$.As in the standard literature,denote: $M(\Gamma)$:={ $\Gamma$ -compatible Beltrami coefficients on $D$ } = {Beltrami differential forms on Riemann surface $X$}= {measurable complex-antilinear bundle automorphims/self-maps of $TX$ of sup. norm $< 1$}.

Here is my question:

(I) Can we have a non-zero $\mu \in M(\Gamma)$ such that $\mu \in C^0(\bar{D})$, or even $C^k(\bar{D})$ ? Put in other words, is there a non-zero element in $M(\Gamma)\cap C^0(\bar{D})$ ? Note that if such a $\mu$ is constant,then obviously $\mu=0$, so the question is same as asking whether there is a non-constant $\mu$.

I was suspecting that in MANY cases there might not be any such $\mu$, because I was thinking the following might be correct, although couldn't prove it, so this could be my (related) second question :

(II) $\text{True or False ?}$ Let $X=D/{\Gamma}$ be a hyperbolic Riemann surface. Let $z\ne w \in D$, fix them. Then there is $\zeta \in S^1 = \partial{D}$ such that there are sequences $\gamma_n\in \Gamma, \nu_n \in \Gamma$ so that $\gamma_n(z)\to \zeta, \nu_n(w)\to \zeta$ as $n\to \infty.$

If (II) is true, then it partially answers (I), but implies that $|\mu|$= constant. Because if there is a non-absolutely constant $\mu \in M(\Gamma)\cap C^0(\bar{D})$, then there are $z\ne w \in D$ such that $|\mu(z)|\ne|\mu(w)|$. Now, as $\gamma_n(z)\to \zeta, \nu_n(w)\to \zeta$ as $n\to \infty,$ and $|\mu(\gamma_n(z))|=|\mu(z)|$, therefore passing to $\zeta, n \to\infty $, we have boundary values of $\mu$ not matching up at $\zeta$.

Any hints/solutions/references will be highly appreciated !

I would be happy if the answer to (I) is yes though, because then we might look at the "smooth" subset of Teichmuller spaces : {restriction of $\mu$-quasiconformal maps fixing $1,-1,i$ to $S^1,\mu \in C^0(\bar{D})$}, and we might consider its contractibility, complex structure etc. :)

Basic Questions about Teichmuller's theorem/quadratic differentials

2011-02-24T04:14:28Z

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will appreciate.

Qn. 1. For considering a minimal dilatation map between open annuli $ A(1,r_1) $ and $ A(1,r_2) $,why don't we consider a certain homotopy class like the general case of Teichmuller's theorem, is it because that all maps between two annuli are homotopic ? Between $f(x) $ and $g(x) $ we construct a homotopy by composition of two homotopies : first radially adjusting $g(x)$ such that $ |f(x)|= |g(x)| $ and then rotating the image of $g(x) $ under the first homotopy along the circle of radius $|f(x)|$ to match with $f(x) $ by the second homotopy ?

Qn. 2. a) Suppose we want two construct a minimal dilatation map between two open ( without boundary ) pair of pants , i.e. 2-sphere minus 3 disjoint closed disks, do we have to look at the minimal dilatation map between certain pair of pants with geodesic boundary of specified boundary lengths which should correspond to the open PPs such that the extremal length of certain curve families ( probably the ones joining the boundaries ) remains invariant by this open surface-to-compact-surface-with-boundary transition ? [ I am just trying to imitate the arguments for the case of annuli ]

b) If the answer to a) is yes, then intuitively can one guess that the minimal dilatation map between the "corresponding" pair of pants with geodesic boundaries ( say the first has longer boundaries: so "fat" PP , the other has shorter boundaries , so it is "thin" PP ) is obtained by "stretching the fat PP along " the common orthogonal to geodesic boundaries ? Is there a more explicit way , like the annuli, to describe this minimal dilation map ?

Qn. 3. I accept the result without proof ( by using Riemann-Roch , mentioned F. Gardiner's book Teichmuller Theory and Quadratic Differentials, P.26, Ch. 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n $ . Now for open annulus $A$, $ g=0, m=2, n=0 $, we get $ dim_RQD(X)=0 $ ! I am a bit puzzled why it is zero ! What should be the genus of an open annulus in any case, shouldn't it be zero ? And for q.diffs on annulus $A$, should we look at $ q=\phi(z)dz^2 $ when $\phi$ is a function on the annulus embedded in complex plane or should we lift it to upper half plane and consider the $ \phi(z) $ with $ \phi(z) = \phi(\gamma(z))\frac{\bar{\gamma'(z)}} {\gamma'(z)}$ for all $\gamma \in Deck(H/A) $ ? I guess the second approach makes more sense ?

Qn. 4. $ FOR REFERENCES :$ I recently finished studying Lipman Ber's paper " Q.C. maps and Teichmuller's theorem " , which proves the uniqueness and existence for closed Riemann surfaces. Is there any good references/books/research paper about Teichmuller's theorem for punctured ( with cusps ) and open Riemann surfaces ?

I am sorry for this very long question if that bothers you.

Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

2012-03-07T21:26:01Z

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :

By $ \bar{P} $ , we denote a topological pair of pants ( that is, a 2-sphere with three open disks removed ) with boundary, and by $P$, we would mean the same without any boundary. Let $\mathcal {Mod(\bar{P})}$ denote the space of all complete hyperbolic metric on $ \bar{P} $ which make the boundary components of $P$ geodesics ; call these geodesics $\gamma_1, \gamma_2, \gamma_3 $ . Since these metrics are completely determined by the hyperbolic lengths $ l_1, l_2, l_3 $ of the three boundary components, we can give $ \mathcal { Mod(\bar{P} ) } $ the co-ordinates $ l_1, l_2, l_3 $ and so think of $ \mathcal { Mod(\bar{P} ) } $ as $\mathbb{R_+}^3$. Consider the following map $ F : \mathcal {Mod(\bar{P})} \cong \mathbb{R_+}^3 \to \mathbb{R_+}^3 $ defined as follows :

$m_i$ is the module of the maximal ring domain in $ \bar{P} $ whose core curve is homotopic to $ \gamma_i $, i.e. for all ring domain $R_i$'s whose core curve is homotopic to $ \gamma_i $, we have $ m_i = sup mod (R_i) $ and this supremum is achieved by a ring domain, which is the the pair of pants $\bar{P}$ minus the unique geodesic joining $ \gamma_{i+1 modulo 3}, \gamma_ {i+2 modulo 3} . $. Now define the map : $ F : \mathcal {Mod(\bar{P})} \to \mathbb{R_+}^3 $ by $ F(l_1,l_2,l_3) = (m_1,m_2,m_3) $. I was wondering whether this map is a homeomoprhism of $ \mathbb{R_+}^3 $. I guess constructing $F$ explicitly would be really hard. Is there any literature along this line ?

( finite ) Blaschke product in higher dimensions ?

2012-05-28T21:51:59Z

Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether there are similar definitions of (finite) Blaschke products in higher ( real ) dimensions,in $\mathbb{R}^n$, $n \geq 3$.

I think, to construct Blaschke product in higher dimensions, we need to keep in mind that $P$ maps $\mathbb{B}^n$ to itself, and $|P(x)|\to 1$ as $ |x| \to 1 $ and $P(\frac{1}{\bar{z}})= \frac{1}{\bar{P(z)}}$.

I myself was trying to define a product of two vectors in $\mathbb{R}^3$ by defining the map using the spherical polar co-ordinates : $P(v,w)= P (v=(r_1,\theta_1,\phi_1), w=(r_2,\theta_2, \phi_2))= (r_1.r_2, \theta_1+\theta_2, \phi_1+\phi_2 )$, resembling the multiplication of complex numbers. But then the question becomes, in order to define Blachke product of say at least two maps, what kind of maps we should really multiply. There is no concept of holomorphic maps on $\mathbb{R}^3$, but we can try to replace them by conformal automorphism of $\mathbb{B}^n$, keeping in mind that $ \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$ are conformal automorphisms of $ \mathbb{D}= \mathbb{B}^2$.

Before proceeding more, I was checking with the math community whether this is the standard way to define higher dimensional Blaschke products, or there are other standard way(s) to define them.Please let me know or cite any reference(s) you know. Thanks !

What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

2012-05-10T22:39:36Z

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with punctures on it, or torus with a cusp attached to it . They could also be torus torus with one or more than one handles attached to it.

For example, see the diagrams on : http://www.maths.bris.ac.uk/~mazag/hyperbolic/index.html

Or see the diagrams on : http://lamington.wordpress.com/2010/04/18/hyperbolic-geometry-notes-4-fenchel-nielsen-coordinates/

to get ideas about what surfaces I am talking about. They are not given by any easy equations.

Is there a software I can use to draw them ? People who study Riemann Surfaces or Hyperbolic Geometry or Teichmmuller Theory would definitely know exactly what surfaces I am talking about.Please let me know if you use such a software. Thanks a lot in advance !!

Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

2010-07-08T20:50:56Z

Hello,

Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also mention some papers if you know.

Thank you !

Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $

2012-04-21T20:58:55Z

The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ?

For detail :

Fix $p \geq 1. $ By the word 'smooth', I will always mean $C^{\infty}$ .Let $U, V$ be two bounded, connected open domains in $ \mathbb{R}^n $ with smooth boundary, so that $\bar{U}\subset V $. Let $u \in W^{1,p}(U) \cap C^0(\bar{U}) $. In many PDE books ( for example , L.C. Evans' "Partial Differential Equations", P. 254 ) there are standard methods described to produce an extension $Eu$ of $u$ such that :

$Eu \in W^{1,p}(\mathbb{R}^n), support (Eu) \subset V, Eu = u $ a.e.on $U$ . The method is first to treat the case where $u \in W^{1,p}(U) \cap C^0(\bar{U}) \cap C^{\infty}(\bar{U}) $ ,locally flatten the boundary $ \partial{U} $, extend $u$ by higher order reflection across the boundary, using partition of unity and then approach arbitrary $u \in W^{1,p}(U) $ by $C^{\infty}(\bar{U})$ functions in $W^{1,p}(U)$ -norm. We do not even need $ u \in C^0(\bar{U}) $ in this proof .

I understand that if $u \in C^{\infty}(\bar{U}) $as well. , then $Eu \in C^\infty(\mathbb{R}^n)$, but is the same true if $u \in C^{\infty}(U) \cap C^0(\bar{U}) $ only, not $C^{\infty}(\bar{U})$ ? This is my main question.

Some basic questions about the proof of Teichmuller's uniqueness theorem

2011-01-17T20:43:21Z

Hello ,

I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I am new to the subject. It would be great if you answer some of my questions :

Here are some questions I might like to ask :

Let $X$ be a closed Riemann surface, $ S_g $ be topological closed oriented genus g surface, $q_X $ be a holomorphic quadratic differential on X.

Why [($X$, $\phi $)] should be an element in $Teich(S_g$) ? ( P. 291 of Farb-Margalit ) . I mean why should we put phi there ? Different phi just gives different laminations of X , so why should we put it there ?
What exactly is meant by " projective classes of $ q_X $" ? ( P 291 of Farb-Margalit ) Why does it give a tangent direction in the tangent space $T_X(Teich(S_g))$ of $Teich(S_g)$ at $X$ ?
( P. 292 of Farb-Margalit ) Given $X, q_X, K > 1$, we can cook up a new closed Riemann surface $Y$ such that there is a Teichmller map f with initial QD $q_X$, terminal QD $q_Y$, and stretch factor K > 1 in the following way :

First puncture $X$ at the zeros of $ q_X$, take natural coordinate chart , and then compose with the affine map f (x,y) = ($ \sqrt(K)x, \sqrt(1/K)y $). But then the new transition maps become $f o (z_1) o(z_2inverse )o f inverse $[ sorry about bad notation ], where o means composition . This new map is NOT holomorphic, although the rest except the f-parts is holomorphic . So how do we get a Riemann surface structure ?

How exactly can we think of a teichmuller map as a map from $QD(X)$, space of holomorphic quadratic differentials on $X$, to $ Teich(S_g) $ ?
1. Finally, what is/are really good reference for this topic ?

The version of Montel's theorem used in the proof of Jenkins-Strebel differential

2012-01-30T22:28:30Z

Hello,

I am afraid that my main question might be a bit too elementary, but still I ask :

In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an open subset $U$ to a hyperbolic Riemann surface $X$ whose range does not lie in a single co-ordinate chart of X ?" Here is how I got to think about the version :

I was reading up the proof of jenkins-Strebel quadratic differential of prescribed heights $b_i$ on a Riemann surface $X$ from kurt Strebel's book " Quadratic Differentials ", Chapter VI, section 21, page 108. Let us, for simplicity, use the version of the theorem in the case of one single Jordan curve $\gamma$. So, in the beginning of the proof, they consider a sequence of ring domains $R_n \subset X $ of homotopy type $\gamma $ with moduli $M_n$ such that $M_n \to M = sup_{n \ge 1 }M_n, 0 < M < \infty $. Then they consider a sequence of conformal maps $g_n : A_n \to R_n $, where $ A_n= { z \in \mathbb{C} : r_n < |z| < 1 } $, where $r_n \to 0 $.

Then they claim that : $ g_n : A_n \to R_n\subset X $ form a normal family. I understand that somehow they are trying to use Montels theorem from complex analysis : a family of holomorphic functions on a domain $U$ in $\mathbb{C}$ which omits two points is normal. But how exactly does this theorem apply to our present case, where the codomain of $g_n$ is a Riemann surface, not a subset of $\mathbb{C}$, so that we can NOT even talk about the absolute value symbol $ | g_n - g_m | $ , so what is the meaning of uniform convergence of a family of holomorphic maps which take values into a Riemann surface $X$ ? And what is the meaning of normality ?

I was thinking of ideas like lifting the maps $g_n $ first to the universal cover $D$ of $X$ with the normalization $g_n(p) =0 \forall n $ for some fixed $p \in A_1 $. Then we can apply Momtels's theorem to the lifted maps since all of them are missing three points, the co-domain being $D$, but then I needed to prove that the projection map $\pi: D\to X $ sends normal family to the normal family, which I was unable to prove.

This is my main question. A detailed explanations would be appreciated ! Thanks !

A question related to the difference between non-tangential and tangential convergence

2011-12-15T22:11:45Z

Let $D$ be the unit open disk in $C$ and $S^1$ be its boundary.

Let me start by defining $z_n\in D\to 1 \in S^1$ "non-tangentially" ( abbreviated n.t. ) as $z_n\to 1 $ but all $ z_n $ lie in an angular domain $A$ at $1\in S^1$, i.e. $A$ is the set of all $z\in D : | Arg ( 1 - z ) | \le \theta $ for some $\theta > 0$. You can think of $A$ as the domain bounded in $D$ by the two straight lines that make an angle $\theta $ and $-\theta$ with the radial line segment $[0,1]$.

Let $z_n = x_n + i y_n$. It is easy to show that there is$ M > 0 $ such that the hyperbolic distance $d_n = d_D(z_n,x_n) \le M \forall n \ge 1 $. The proof is to write $e^{d_n} = \frac{1+|\frac{z_n - x_n}{1 - x_n. z_n}|}{1-|\frac{z_n - x_n}{1 - x_n. z_n} |}$. Call $a_n = |\frac{z_n - x_n}{1 - x_n. z_n}| = \frac{|y_n|}{|1- x_n.z_n|}$ .Note that $ |1 - x_n. z_n|\ge |1- z_n|$ when $x_n > 0 $, therefore $ a_n \le \frac{|y_n|}{|1- z_n|} = sin (\theta_n) \le k < 1 \forall n $,here $\theta_n$= angle between $[1,z_n]$ and $[0,1]$. Hence $e^{d_n} \le \frac{1+k}{1-k}$ and it is proved.

But my question is : if $z_n\in D\to 1 \in S^1$ tangentially, i.e. NOT lying in ANY such angular domain $A$ as mentioned before, must we have $d_n = d_D(z_n,x_n)\to \infty$ as $n\to \infty $ ? In other words, must

$ a_n = |\frac{z_n - x_n}{1 - x_n. z_n}| = \frac{|y_n|}{|1- x_n.z_n|} \to 1 $ Note that to prove the non-tangential case, we used the inequality $ |1 - x_n. z_n|\ge |1- z_n|$ in a suitable way for our favor, here we cannot do that .

If the limit is not infinity, can we have some tangential convergence for some sequence $z_n \to 1$ such that $d_n = d_D(z_n,x_n) \le M \forall n \ge 1 $ ?? Can we have explicit example or characterization of such tangential convergences ?

Boundary regularity of the solution to the Beltrami equation

2011-10-28T00:26:32Z

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:

Let us consider the orientation-preserving homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z, ||\mu||_{L^{\infty}(D)}\le k\le 1, D$ is the unit disk in the complex plane $C$, $f_z, f_\bar{z} $ are the partial derivatives of $f$ w.r.t $z, \bar{z} $ respectively. I am looking for the known results on the sufficient , or, necessary and sufficient condition on the Beltrami coefficient $\mu$ such that any solution of the equation is at least $C^1(\bar{D})$, but I would be happier to get $C^k(\bar{D}), 1\le k \le \infty $, or even real-analytic upto the boundary. By $C^k(\bar{D})$, I mean the $k$th derivative exists and is continuous upto the boundary. Holder continuity of solutions is also I would be interested in. ( Please note that I stated 'any' solution , since we know that any two solutions differ by a Mobius transformation of $D$.

In general, I am looking for results for the boundary regularity of the solution to the Beltrami equation.

I would highly appreciate if you state any known results on this topic , thanks in advance !

How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

2011-11-03T14:42:56Z

Let me first state the definitions :

A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ; $ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi_1(X,c(1)),$ the homotopy class $[c]$ can NOT be written as $[c]= [\gamma]^n$ for some closed curve $\gamma$ with $\gamma(0)=\gamma(1)=c(0)=c(1) $ and for some $n\ge 2$.

My question is : rigorously prove that simple closed curves are primitive .

It is visually pretty clear , but I have difficulty proving it. Thanks !

A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

2011-09-27T16:24:23Z

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or suggest a better reference.

Let $\Omega$ be defined by the open domain in $\mathbb{H} $ bounded by the lines $\Re(\tau)=0,\Re(\tau)=1, $ and the circle $|z-1/2|=1/2 $. ( $\Re$ means real part. )

In Ahlfors's complex analysis, second edition, page 272, Ahlfors proves that the modular function $\lambda $ maps $\Omega $ ( which is the open right half of the fundamental domain of the congruence subgroup modulo 2 group $\Gamma(2)$ ) conformally onto the upper half plane $\mathbb{H} $ ( which, while combined with the fact $\lambda o \phi = \lambda \forall \phi \in \Gamma(2) $ and that $\Omega \cup \Omega^* $$\cup $ {positive y-axis} is a fundamental domain for $\Gamma(2)$,( $\Omega^*$ is the reflection of $\Omega $ in the positive y- axis ) and that $\lambda$ is surjective,proves that $\lambda$ is a covering space for $\mathbb{C}\backslash{0,1}$.

I have some questions regarding the proof of covering space :

How exactly do we prove that $\lambda : \mathbb{H} \to \mathbb{C}\backslash {0,1} $ is surjective ? I think this should follow from the my queston # 2.
I am also unable to follow Ahlfors's argument on the first paragraph of P. 273, apparently which seems rather sketchy to me ( see P. 272 ) that $ \forall w_0\in \mathbb{H}, \frac{1}{2\pi i}\int_\Gamma \frac{\lambda'(\tau)}{\lambda(\tau)- \ w_0}d\tau = 1 $ and $ \forall w_0\in \mathbb{H^*}, $ ( the lower half plane )$ \frac{1}{2\pi i}\int_\Gamma \frac{\lambda'(\tau)}{\lambda(\tau) -\ w_0}d\tau = 0 $ , where $\Gamma$ is obtained by taking the boundary of the truncated region bounded by $\Re(\tau)= t_0 > 0,\Im (\tau) =0, \Im(\tau)=1$ where $t_0$ is sufficiently large, and two sufficiently small circles tangent to the x-axis at 0 and 1 respectively. This will prove, by argument principle, that $\lambda$ takes each $ w_0 \in \mathbb{H} $ exactly once in $\Omega$. But why are the integrals 1 and 0 in the two above cases ? Could you please explain that in more detail ? Or suggest a better reference that is easier to follow ?

Thanks very much !

Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

2011-09-04T16:59:10Z

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f :H^m \to H^n $ w.r.t the Hyperbolic metric in section 1, "Estimates on Douady-Earle extension" by using the connection forms as follows: (for me, I am happy with $m=n=2$, so I am just re-writing everything in $2$-dimensions). Let $ \theta_i, \theta_{ij} $ respectively denote the local orthonormal coframe on $H^2$ (or $D^2$) and its corresponding matrix of connection forms, $1\le i,j,\le 2$. For a map $\Phi: D^2 \to D^2$, one defines the energy density of $\Phi$ by $ e(\Phi)= \sum_{i,\alpha} {(f_{i}^{\alpha}})^2$, where $f_{i}^{\alpha}$'s are given by $F^*(\theta_\alpha)= \sum_{i}f_{i}^{\alpha}\theta_i $. Then according to this paper, the Hessian of $\Phi$ is denoted by $f_{ij}^{\alpha}$, is defined by :

$\sum_{j} f_{ij}^{\alpha}. \theta_j = df_i^{\alpha} - \sum_j{f_j^\alpha}.\theta_{ij} + \sum_{\beta}f_{i}^{\beta}F^*(\omega_ {\beta} {\alpha} ) $

and they write the norm of the Hessian w.r.t. this orthonormal coframe as :$|Hess(f)|^2=\sum_{ij\alpha} (f_{ij}^{\alpha})^2 $.

I guess I am a little confused by this definition. So my questions are :

1) Previously I knew ( from Riemmanian Geometry textbook of Peter Petersen or John Lee P.54 ) that $Hess(\Phi)=\nabla^2(\Phi) $ is a $(2,0)$ tensor defined by $Hess(\Phi)(X,Y)= g( \nabla_X(\nabla\Phi),Y) $, where $g$ denotes the hyperbolic metric on the open unit disk $D^2$.

Is the new definition of the paper the same as the old definition ? And what should be the norm according to the old definition ?

2) The proposition 1.2, (iii) states that under certain condition, the norm of the Hessian ( as defined before ) of $\Phi$ is $ < \epsilon $ .Does it mean that we can say the standard partial derivatives of $\Phi$, i.e. $\Phi_{xx}, \Phi{xy}, \Phi{yy} $ have norms less than $ < 4\epsilon ( 1- |z^2| )^2$ at the point $z$ under the same condition ( that condition is not necessary for my question here ) , given that we are working with $D$ with the standard Poincare metric on it ,i.e., $g= 4\frac{|dz|^2}{( 1- |z^2| )^2}$?

Thanks so much !

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

2011-09-01T15:58:07Z

Hello,

This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :

If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p > 2 $ is fixed and sufficiently close to 2 ( because of Calderon-Zygmund theorem used in in the proof of the measurable Riemann Mapping Theorem ), $ L^{\infty} $ norm of $ \mu \le \frac{k-1}{k+1}, k> 0 $ then the solution to the Beltrami equation $f_\bar{z}= \mu . f_z$ is a $\mathcal{C^1}$ diffeomorphism of the complex plane $ \mathbb{C} $.

The above lemma is part of the proof of the measurable Riemann mapping theorem, which generalizes the above lemma with $\mu \in L^\infty(\mathbb{C})$.

I have two questions regarding the above lemma :

1) I went through the proof of the lemma, and it seems like we don't need $\mu$ to be compactly supported.Do we ? [ The reason I am asking is the theorem 1 on page 54 is assumes $\mu$ to be compactly supported and they really use it. Also,you look some other book, e.g. : Imayosgi-Taniguchi's Techmuller Theory , chapter 4, Theorem 4.25, they use the phrase "under the same circumstances of theorem 4.24", which mean they want $\mu$ to be compactly supported, which I don't see why ]

2) Is the following true ? ( I am basically replacing the domain $\mathbb{C}$ by $\mathbb{D}$)

If $\mu:\mathbb{D}\to \mathbb{D} \in W^{1,p}(\mathbb{D}), p > 2 $ and also real-analytic in $\mathbb{D}$, then any solution to the Beltrami equation $f_\bar{z}= \mu . f_z$ is a $\mathcal{C^1}$ diffeomorphism of the open unit disk $ \mathbb{D} $ and also $ f \in C^1({\bar{\mathbb{D} }}) $ [ the last condition is crucial ].

I was trying to answer question no 2 by assuming the following, which might not be correct :

If $\mu \in W^{1,p}(\mathbb{D})$ , then it is Holder continuous on ${\mathbb{D}}$ with Holder exponent $1- \frac{2}{p} $ ( well, I am not assuming this,it follows from the theory of Sobolev Spaces, see Evans' PDE book for example ) and hence is uniformly continuous on $\bar{\mathbb{D} }$.

( probably correct/incorrect ) assumption : Then there exists $ g\in C^\infty(D_2)$ such that $g|_{S^1}= \mu | _{S^1},g \in W^{1,3}(D_2) $ . $D_2$ denotes the ball with radius $2$, centered at $0$.

The reason I was doing this is to transfer the problem to a Beltrami equation on $ \mathbb{C} $, by extending the Beltrami coefficient from $\mathbb{D}$ to $\mathbb{C}$ and the reason I want $L^3$ is that for a finite measure space ( balls of finite radius ) $ L^3 \subset L^p \forall p \le 3 $

Any hints or suggestions or detailed answers for question # 2 ? Thank you !

Questions about Douady-Earle/ conformally natural extensions ( from Douady-Earle's paper )

2011-08-28T17:33:40Z

In the paper " Conformally natural extension of the homeomorphisms of the circle " by Adrien Douady and Clifford Earle ",they showed that for any orientation preserving homeomorphism $\phi$ of $S^1$,there is an extension $\Phi= E(\phi)$ to the open unit disk $D$ of $\phi$ , which is a real analytic diffeomorphism of $D$ and a homeomorphism of the closed unit disk $\bar{D}$. The extension operator $E$ has the property that $E(\alpha o \phi o \beta ) = \alpha o E(\phi) o \beta \forall \alpha, \beta \in Aut(D)$. $Aut(D)$ denotes the set of all conformal automorphisms of the open unit disk $D$, each of which has a homemorphic extension to the boundary. This property is called 'conformal naturality'. The proof of $E$ is partially constructive, but kind of detailed. See the paper here :

http://www.springerlink.com/content/hw88761257310165/

I have two questions are from this paper :

(1) On P. 31, section 5 ( Quasiconformal extensions ) of the paper, they define the two positive functions $\alpha(\phi), \beta(\phi) $ on $D$ and they state that : the Lipchitz continuity of $ \Phi, \Phi^{-1}$ with respect to the Poincare metric is equaivalent to having two positive numbers $a, b $ such that :

$ a\leq \alpha(\phi) \leq \beta(\phi) \leq b \forall z \in D $ (5.1)

I need to understand what exactly they mean here by the Lipchitz continuity of $ \Phi, \Phi^{-1}$ with respect to the Poincare metric and why is that equaivalent to having two positive numbers $a, b $ such that ( 5.1 ) holds. The proof of (5.1) for $k$-quasisymmetric homeomorphisms of $S^1$ fixing three points on $S^1$ ,however, is given in the paper.

(2) Apart from the case where the circle homeomorphism $\phi$ is the restriction of a Mobius transformation on $S^1$, is there any case, i.e. any regularity condition on $\phi$ where we can have the $k$-th order ( I would be happy with $k\leq 2 $ ) partial derivatives w.r.t $z, \bar{z}$ of the Douady-Earle extension is bounded ,may be by a constant $C(\phi) $ depending on $\phi$ on the whole open unit disk $D$ ? Of course, it would be even better to have $C(\phi)$ depend only on the quasisymmetric constant of $\phi$ !

Comment by Analysis Now

2013-03-02T21:26:03Z

Thank you very much!

Comment by Analysis Now

2013-02-26T04:47:57Z

Severe mistake: I changed the question: it MUST have been $C^{k,\alpha}$. Thanks!

Comment by Analysis Now

2012-12-17T17:48:58Z

quid: that IS a productive answer and I was absolutely not aware of it. yes, Juergen Jost has books on Riemann surfaces and his research areas are very connected to Teichmueller theory, so it does make sense, and thanks a lot for your help, I sincerely appreciate it!

Comment by Analysis Now

2012-12-17T15:25:15Z

quid: if you go to Uppsala Mathematics department, click on employees, many times you cannot find direct link to their personal webpages, but whne you google their names, you can find it eventually. I double-checked that, and you feel free to double-check that as well. Also, there is no "research groups" in many European Universities, and personal webpages of faculty are not in English many times!

Comment by Analysis Now

2012-12-17T01:53:00Z

To Prof. Eremenko: of course I did all the search, and I know some people working in the area (my question itself clearly indicates that), my goal was to not miss any important researcher. To Prof. Webster: my research areas are Riemann surfaces, in the broader sense, and I mentioned all the particular subject areas related to it, so that people who work in these areas might know people/universities in Europe and might be kind enough to mention them. And yes, European system of posting jobs needs to be improved!

Comment by Analysis Now

2012-12-17T01:46:30Z

(contd.) webpages !!(try Uppsala Univ. for example), so my search faced many obstacles, so I took this way to ask in here, but some people voted to close it. I thank you for your comments again. People who closed this question despite my starting remark, I honestly feel very disappointed at your action, because it is related to my job-search and not everybody has a easy way to find the people they want to work with!!

Comment by Analysis Now

2012-12-17T01:42:30Z

Thanks for your comments! I did talk to my advisor, he doesn't know much about European mathematicians working in those topics, I also did all the search I could: going to different conference webpage finding European attendees, searching for research papers and searching the Europeans authors, going to the European mathematics departments and trying to find people, but despite all these efforts, I couldn't really find enough people, unlike the US. Also, in European math. dept.s' websites are not so well-organized and they are relative;ly smaller(sometimes you can't even see the faculty

Comment by Analysis Now

2012-11-13T04:04:01Z

To Pietro, thanks. If for the first question, I JUST need $C^{1,\alpha}-regularity$ near $1$, i.e. if I want to get $f'(a)-f'(1)=O(|a-1|)^{\alpha},$ can we get it from the alternate definition. I am unable to fill out the details.

Comment by Analysis Now

2012-11-13T03:59:15Z

(To Conor, continued): every point $\zeta \in S^1,$ I have $F(z)-F(\zeta)-l_{\zeta}=O(|z-\zeta|^{1+\alpha})$, where the constant of Holder continuity is locally uniform ? My second comment, passing from difference quotient to the derivative expresses my main concern. You can ignore the rest.

Comment by Analysis Now

2012-11-13T03:54:47Z

(To Conor, continued): $|f(a)-f(1)-f'(1)(a-1)|=O(|a-1|^{1+\alpha})$, which gives $\frac{f(a)-f(1)}{a-1}-f'(1)=O(|a-1|^{\alpha})$. From here, how do you get $|f'(a)-f'(1)|=O(|a-1|^{\alpha})$? I am tempted to use mean-value theorem, but that is not giving me the answer! About the locally $C^{1,\alpha}$ versus globally $C^1{1,\alpha}$, I think circle $S^1$ being compact, they are the same. Anyway, for my question 2, there was a mistake in my question; I know that my $F$ is $C^2(\mathbb{D})$, so if I try to prove $F\in C^{1,\alpha}(\mathbb{D})$, then isn't it enough that I prove: locally near

Comment by Analysis Now

2012-11-13T03:43:13Z

To Conor: first, thanks for your answer. I think in your answer, $l_x= u(0) + Du_x ?$ I am still a little doubtful about how I can pass from the difference between the difference quotient and the linear approximation to the difference quotient between the derivatives between these two points. If I use my notation, then: $r=|a-1|$, and here I am also working with a domain with boundary(which would not probably make any difference in this case):

Comment by Analysis Now

2012-11-12T18:02:48Z

To Paul Garrett: do you think the employers will take the papers seriously if I refer to them in my cover letter as being enlisted under my homepage (and indeed put them in my homepage, but not on arXiv?) I do understand your point though, that putting paper on arXiv is MORE serious than putting on webpage, because I am willing to risk embarassment by doing so.

Comment by Analysis Now

2012-11-12T17:56:22Z

Thanks for all your kind comments and answers.

Comment by Analysis Now

2012-11-06T03:51:44Z

Thanks you Lasse Rempe-Gillen. I will take a look at it.

Comment by Analysis Now

2012-10-26T07:00:46Z

Thanks for the comment, but in my website, I will give the downloadable link to them.