Conformal mappings and isometric immersions under second order Sobolev regularity

Liu, Zhuomin (2013) Conformal mappings and isometric immersions under second order Sobolev regularity. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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 $\bbbr^{n+1}$ (co-dimension 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 $p\geq (n-2)/4$. This is optimal in the sense that if $|Df|^p\in W_{\rm{loc}}^{1,2}$ for $p< (n-2)/4$, it may not be a M\"{o}bius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec-Martin 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Status: Unpublished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:
Liu, Zhuominzhl26@pitt.eduZHL26
ETD Committee:
Committee ChairHajlasz, Piotrhajlasz@pitt.eduHAJLASZ
Committee MemberManfredi, Juanmanfredi@pitt.eduMANFREDI
Committee MemberLewicka, Martalewicka@pitt.eduLEWICKA
Committee MemberBeatrous, Frankbeatrous@pitt.eduBEATROUS
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