Liu, Zhuomin
(2013)
Conformal mappings and isometric immersions under second order Sobolev regularity.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
We consider two classes of vector valued functions with conformal constraint conformal mappings from an $n$dimensional domain into $\bbbr^n$ and isometric immersions of an $n$dimensional domain into $\bbbr^{n+1}$ (codimension one) for $n\geq 3$.
Iwaniec and Martin proved that in even dimensions $n\geq 3$, $W_{\rm{loc}}^{1,n/2}$ conformal mappings are M\"{o}bius transformations and they conjectured that it should also be true in odd dimensions. In the first part of this manuscript, we prove this theorem for a conformal map $f\in W_{\rm{loc}}^{1,1}$ in dimension $n\geq 3$ under one additional assumption that the norm of the first order derivative $Df$ satisfies $Df^p\in W_{\rm{loc}}^{1,2}$ for $p\geq (n2)/4$. This is optimal in the sense that if $Df^p\in W_{\rm{loc}}^{1,2}$ for $p< (n2)/4$, it may not be a M\"{o}bius transform. This result shows the necessity of the Sobolev exponent in the IwaniecMartin conjecture.
In the second part, we prove the developability and $C_{\rm loc}^{1,1/2}$ regularity of $W^{2,2}$
isometric immersions of $n$dimensional domains into $\bbbr^{n+1}$ for $n\geq 3$. The result is sharp in the sense that $W^{1,p}, 1\leq p\leq \infty$ and $W^{2,p}, 1\leq p<2$ isometric immersions may not be developable. Based on this result, we also prove that if the domain is $C^1$ and convex, smooth isometric immersions are strongly dense in this space.
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Details
Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
29 January 2013 
Date Type: 
Publication 
Defense Date: 
30 November 2012 
Approval Date: 
29 January 2013 
Submission Date: 
6 December 2012 
Access Restriction: 
No restriction; Release the ETD for access worldwide immediately. 
Number of Pages: 
157 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Dietrich School of Arts and Sciences > Mathematics 
Degree: 
PhD  Doctor of Philosophy 
Thesis Type: 
Doctoral Dissertation 
Refereed: 
Yes 
Uncontrolled Keywords: 
Conformal mappings, isometric immersions, Sobolev Spaces. 
Date Deposited: 
29 Jan 2013 21:39 
Last Modified: 
15 Nov 2016 14:08 
URI: 
http://dscholarship.pitt.edu/id/eprint/16801 
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