1,720,964 research outputs found
Extension domains.
No full textThis research is concerned with a problem in higher dimensional quasiconformal mappings and a related problem in the complex plane. Specifically, we study domains D in \IR\sp{\rm n} for which each quasiconformal self map of D extends to a quasiconformal self map of \IR\sp{\rm n}. For n = 2, it was shown by Ahlfors and Rickman that a simply connected domain D has the above extension property if and only if D is a quasidisk. Suppose that G is a Jordan domain in \overline{\IR}\sp2. We consider the relationship between the geometry of G and the above extension property for D = G \IR. Vaisala showed that G \IR is quasiconformally equivalent B\sp3 if and only if G is an inner chordarc domain. For our extension problem we need to consider the geometry of such domains. We show that a Jordan domain \rm G\subset\IR\sb2 with locally rectifiable boundary is an inner chordarc domain if and only if for each straight cross-cut = (z, w) of G, where is the shorter arc in G joining z, w. Next G is an inner chordarc domain if and only if G is a John domain and G is regular. Finally, a characterization in terms of the corresponding Riemann map is given. Our extension results are the following. If G is a bounded inner chordarc Jordan domain in \IR\sp2, then G \IR does not have the extension property for the class of quasiconformal maps fixing . On the other hand, if G is an inner chordarc domain and if G* = \IR\sp2\\G, then G* \IR has the extension property for quasisymmetric maps. Next if D = G \IR is quasiconformally equivalent to B\sp{3}, then D has the quasiconformal extension property if and only if D is a quasiball. Finally, using the reflection properties of quasiconformal maps we show that if G is an unbounded quasidisk in \IR\sp2, then G \IR has the quasiconformal extension property in \overline{\IR}\sp3. We also construct a bounded quasidisk G such that G \IR has the above property.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105185/1/9116182.pdfDescription of 9116182.pdf : Restricted to UM users only
Univalence criteria for analytic functions.
No full textThis thesis is devoted to the study of univalence criteria for analytic functions, in particular a domain constant known as the inner radius of D. Let D be a simply connected plane domain and let B be the unit disk. We define the inner radius of D, by\sigma(D) = \sup\{a : a \geq 0,\ \Vert S\sb{f}\Vert\sb{D}\ \leq\ a\ {\rm implies}\ {\it f\/}\ {\rm is\ univalent\ in}\ D\}.Here S\sb{f} is the Schwarzian derivative of f, \rho\sb{D} the hyperbolic density on D and\Vert S\sb{f}\Vert\sb{D} = {\sup\limits\sb{z\in D}}\vert S\sb{f}(z)\vert\rho\sbsp{D}{-2}(z).Domains for which the value of is known include disks, angular sectors and regular polygons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that = 2-\Vert S\sb{h}\Vert\sb{B} where h maps B conformally onto D. Because of the importance of this property, we say that D is a regular domain if = 2 -\Vert S\sb{h}\Vert\sb{B} is satisfied. First we use regularity to give a simple new proof of the result on regular n-sided polygons P\sb{n}. Next we study rectangles and equiangular hexagons. We prove that if R is a rectangle whose ratio of longer over shorter side is bounded from above by a specific constant ( 1.52346 then R is regular and = = \sigma(P\sb4). In a similar fashion, we prove that if H is an equiangular hexagon whose sides form the sequence baabaa with 1.67117 then H is regular and = = \sigma(P\sb6).. An interesting problem is to characterize regular domains. For domains of bounded boundary rotation with convex corners we show that{\limsup\limits\sb{\vert z \vert\to1}}\vert S\sb{h}(z)\vert(1 - \vert z\vert\sp2)\sp{\sp2} = \Vert S\sb{h}\Vert\sb{B}is a sufficient condition for regularity, where h maps B conformally onto D. Some results previously known for B can now be extended for all regular domains, in particular theorems of Gehring-Pommerenke, Ahlfors and Minda. The last part of the thesis is devoted to investigating an alternative domain constant, The only domains for which is known are disks. We demonstrate some bounds on for convex angular sectors.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104352/1/9513432.pdfDescription of 9513432.pdf : Restricted to UM users only
Quasiregular mappings and Royden algebras.
No full textLet be as domain in euclidean n-space and A() the Royden algebra of . Then A() is the Banach algebra of all functions \rm u \in C(\Omega) \cap L\sp\infty (\Omega) \cap L\sbsp{n}{1}(\Omega) with addition and multiplication defined pointwise and norm \rm \Vert u \Vert \sb\Omega = \Vert u \Vert \sb\infty + \Vert \nabla u \Vert \sb{L\sp n}\sb{(\Omega)}. It is known that domains and \Omega\sp\prime are quasiconformally equivalent if and only if there exists an algebra isomorphism T:A( \Omega \sp\prime)\to\rm A(\Omega). In this thesis, we characterize quasiregular mappings f:\Omega\to\Omega\sp\prime with finite multiplicity as exactly those mappings induced by algebra isomorphisms T:A(\Omega\sp\prime)\to\rm A where A is a subalgebra of A() which satisfies certain conditions concerning the separation of sets in by functions in A. We characterize closed quasiregular mappings in a similar fashion and show that the multiplicity of a quasiregular mapping induced by T:A(\Omega \sp\prime )\to \rm A \subset A(\Omega) is bounded above by T\Vert \sp{\rm n \sp2}. We also investigate the maximal ideal space * of A(), which is a compact, Hausdorff, topological space. We let denote the Royden boundary of which is the set *. If two domains are quasiconformally equivalent, then their Royden boundaries must be homeomorphic. Using the theory of nets, we are able to characterize elements of as certain types of nets for which no subnet is a sequence. We define fibers over points in and show that even in the special case that is the unit ball and all fibers over are homeomorphic, is not the natural topological product of with any particular fiber. Finally, we discuss two ways in which A() reflects the geometry of . First, we define a condition on sequences of level sets linked in which guarantees that is not quasiconformally equivalent to the unit ball. Second, we use Sario's and Nakai's definition of the harmonic boundary of to give a simple condition on the Royden algebra which is equivalent to the existence of a Green's function on .PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105449/1/9124109.pdfDescription of 9124109.pdf : Restricted to UM users only
The quasihyperbolic metric, growth, and John domains.
No full textThe research in this thesis stems from three elements of classical function theory. The first is a criterion due to Hardy and Littlewood for a function to be Holder continuous in the unit disk \rm I\!B\subset\doubc. The second is a famous inequality due to Bernstein which bounds the modulus of the derivative of a polynomial in terms of the degree and the L\sp\infty-norm of the polynomial in . The third is a class of domains first considered by Fritz John in his studies of plane elasticity and rigidity of local quasi-isometries. Suppose that f is a function analytic in and that Then the theorem of Hardy and Littlewood mentioned above asserts that\vert f\sp\prime(z)\vert\le m\ {\rm dist}(z, \partial {\rm I\!B})\sp{\alpha-1}for all if and only if\vert f(z\sb1)-f(z\sb2)\vert\le{M\over\alpha}\vert z\sb1-z\sb2\vert\sp\alphafor all z\sb1,\ z\sb2\in\rm I\!B, where m and M depend only on each other. Bernstein's inequality states that if p(z) is a polynomial of degree n, then\sup\sb{\rm I\!B}\vert p\sp\prime(z)\vert\le n\ \sup\sb{\rm I\!B}\vert p(z)\vert.\eqno(1) A domain D\subset\IR\sp{n} is a b-John domain if each pair of points x\sb1,\ x\sb2\in D can be joined by an arc for which\min\sb{j=1,2}l(\gamma(x\sb{j},y))\le b\ {\rm dist}(y, \partial D)for all , where \gamma(x\sb{j},y) is the subarc of with endpoints x\sb{j} and y. A domain is John if it is b-John for some constant b, and a simply-connected John domain in the plane is a John disk. John domains appear naturally in many areas of analysis, including complex dynamics, approximation theory, and elasticity. My research concerns the following four questions. (1) What analogues of the Hardy-Littlewood result hold when ? (2) What analogues of this result hold when is replaced by a domain D\subset\doubc and ? (3) What analogues of this result hold when f is an arbitrary function defined in a domain D \subset \IR\sp{n} and ? (4) For which continua E\subset\doubc does an analogue of inequality (1) hold? John domains arise in examining the second, third, and fourth questions listed above.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104976/1/9624658.pdfDescription of 9624658.pdf : Restricted to UM users only
Properties of John disks.
No full textA domain D in euclidean n-space R\sp n is said to be a John Domain if there exist a point x\sb0 in D and a constant such that point x\sb1 in D can be joined in D to x\sb0 by an arc such that the length of the subarc from x\sb1 to x in is bounded above by c times the distance from x to the boundary of D. This class was first studied by Fritz John in 1961 in connection with his work on elasticity and local quasi-isometries. In this thesis we study John disks, the John domains D in R\sp2 for which is a Jordan curve, and we give new conformally invariant, geometric and function theoretic properties and characterizations for this class. The conformally invariant properties involve harmonic and hyperbolic measure, the geometric conditions concern quasidisks and a quasiextremal distance property, and the function theoretic properties consist of an analogue of the Bernstein inequality and the conformal mapping of the exterior of the unit disk onto the exterior of D. In the final chapter we characterize the compact sets on which the above analogue of the Bernstein inequality holds in terms of the exterior mapping function.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105598/1/9135685.pdfDescription of 9135685.pdf : Restricted to UM users only
The geometry of discrete groups.
No full textThis dissertation is concerned with discrete groups of Mobius transformations. A Mobius transformation is a conformal self mapping of the extended complex plane and a Mobius group is discrete if it does not contain any convergent sequence of distinct elements. We are mainly interested in two generator groups since for essentially all Mobius groups, G is discrete if and only if every two generator subgroup of G is discrete. We obtain discreteness conditions for Mobius groups by considering various iterative commutator sequences. For example, we show that if is discrete and f and g are conjugate, then 2 2 cos(/7) is the sharp lower bound for the distance from the trace of the commutator (f,g\rbrack = fgf\sp{-1}g\sp{-1} to 2. This result gives rise to many geometric constraints for discrete Mobius groups. In the study of the iterative commutator sequences, one makes use of the fact that three trace parameters tr\sp2(f),\ {\rm tr}\sp2(g), and tr (f,g) determine the two generator group up to conjugacy whenever tr (f,g) is not equal to 2. In particular, one can replace g by an elliptic h of order two so that is discrete and tr (f,h) equals tr (f,g). In contrast to the two generator case, we show that the corresponding six trace parameters determine two different conjugacy classes of three generator Mobius groups and that the groups in one conjugacy class can be discrete while the groups in the other conjugacy class are not. These two conjugacy classes coincide if the generators are not parabolic and the axes of two generators intersect orthogonally. We also give an example to show that using an order two element in a three generator discrete group may change the discreteness. The chordal norm d(f) = d(f, id) is the maximum of the chordal distance d(fz,z) over all points z in \bar\IR\sp{n}. It measures the maximum chordal derivation of f from the identity, and d(f) = 2 if and only if f maps one point of a pair of antipodal points of \bar\IR\sp{n} onto the other. By means of the trace inequalities we obtain lower bounds of max for a discrete group in dimension 2 as well as in dimension n.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/104924/1/9624583.pdfDescription of 9624583.pdf : Restricted to UM users only
The apollonian metric, sets of constant width and Moebius modulus of ring domains.
No full textIn this thesis we study the concepts of apollonian metric, sets of constant width and Mobius modulus of a ring domain. The apollonian metric, introduced in 1995 by A. Beardon, can be thought of as a generalization of the hyperbolic metric to more general domains. We give a positive answer to a question of Beardon, namely, that the apollonian isometries are the restrictions of Mobius transformations, in two instances. Sets of constant width have been an object of study by geometers for several centuries; some non-trivial examples of such sets were already known to Euler. We give a new characterization of these sets in terms of the apollonian metric. In particular, we give characterize balls in terms of this metric. We introduce the Mobius modulus of a ring domain in the context of quasiconformal mappings. It shares many properties with the classical conformal modulus of such domains which is one of the main tools in the study of quasiconformal mappings. We solve Teichmuller's problem for a Mobius modulus, and as a result we settle a conjecture of Vuorinen.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132111/2/3057971.pd
Domains with a local to global norm condition.
No full textLet D be any domain and B any ball in Euclidean n-space. For a real valued function u on D and a point x in D, we define the oscillation over D of u at x as the absolute value of the difference of u(x) and the average of u over D. We consider the normalized integral over D of the oscillation of u over D to the p\sp{\rm th} power, where p 1. Here we normalize this integral by dividing by the measure of D and then taking the p\sp{\rm th} root of the entire expression. Call the final result the p\sp{\rm th} average oscillation of u over D. We seek to characterize domains D which have the property that the p\sp{\rm th} average oscillation of u over D is less than a constant multiple of the supremum over all balls B in D of the p\sp{\rm th} average oscillation of u over B. We call domains with this property L\sp{\rm p}-averaging domains. For the case p = 1, the supremum above reduces to the well known bounded mean oscillation norm. We present the following characterization of L\sp{\rm p}-averaging domains. A domain is an L\sp{\rm p}-averaging domain if and only if the quasihyperbolic metric raised to the p\sp{\rm th} power is integrable in D. Next we examine properties and examples of L\sp{\rm p}-averaging domains. In particular, we prove that L\sp{\rm p}-averaging domains satisfy a Poincare inequality, are preserved by quasi-isometries but not quasiconformal maps, and include John domains as a subclass. Lastly we consider an analogous question for the maximal oscillation. Here we define the maximal oscillation over D of u as the supremum over all x and y in D of u(x)u(y). Now we study domains D with the property that the maximal oscillation of u over D is bounded by a constant multiple of the supremum over all balls B in D of the maximal oscillation of u over B. We call such a domain an oscillation domain and provide certain necessary and sufficient geometric conditions for a domain to have this property.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128240/2/8821657.pd
Domain constants of injectivity.
No full textLet f be a locally injective mapping of a simply connected hyperbolic domain D into \bar\doubc. Under what circumstances is f injective? We assign norms to f in such a way that if f is sufficiently small, then f is injective. Using these norms we associate domain constants of injectivity to D. We consider the cases where f is meromorphic and locally K-quasiconformal. For f meromorphic let S\sb{\rm f} denote the Schwarzian derivative and let \rho\sb{\rm D} denote the density of the Poincare metric in D. The inner radius of univalence (D) is defined as the supremum of the numbers a 0 such that \vert\rm S\sb{f}(z)\vert a\rho\rm\sb{D}(z)\sp2 for all z D is a sufficient condition for f to be injective. We define normal circular triangles and show that (D) = 2k\sp2 if D is a normal circular triangle whose smallest angle is k. Using this result we show that if D is a regular n-sided polygon, then (D) = 2 . For locally K-quasiconformal mappings we define (D,K) as the supremum of the numbers b with the property that if log J\sb{\rm f}\Vert\sb\* \leq b, then f is injective; when no such constants b exist, we set (D,K) = 0. Here \Vert\cdot\Vert\sb\* denotes the BMO norm in D. Then we define K(D) as the supremum of the numbers K for which (D,K) 0. It is known that K(D) 1 if and only if D is a quasidisk, and that K(D) 2 for all D, with equality if D is a disk. We prove that K(D) = 2 also when D = z: z 1, Re (z) cos(k) for k 1/7. For this we show that D has a length-area property which we call the crosscut property. We prove also that domains with this property are convex and are K-quasidisks where K is bounded by an absolute constant.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128207/2/8821554.pd
Quasiextremal distance domains and quasiconformal reflections.
No full textQuasiextremal distance (or QED) domains, domains whose complements do not change the extremal distance between continua by more than a fixed factor, were introduced by Gehring and Martio in connection with the theory of quasiconformal mappings in the Euclidean n-space \IR\sp{\rm n}. The purpose of this thesis is to study the geometry of these domains. With each domain D we associate two QED constants M(D) and M*(D), which measure how far D is from being a Mobius ball. It is known that M(D) and M*(D) are Mobius invariant and that M(D)=M*(D) = 2 when D is a ball or half space. We characterize all domains D for which M*(D) = 2 and prove that M*(D) = 2 if and only if D is a Mobius ball minus an NED set, a set each compact subset of which is the complement of a QED domain and has Lebesgue measure zero. To prove this, we establish some results about the extremal functions for the capacities of condensers by using variational integral methods. These functions turn out to be p-harmonic functions which are weak solutions of the non-linear equation div(u\vert\sp{\rm p-2}\nablau) = 0. We estimate the constants M(D) and M*(D) for different kinds of domains. For any domain D M(D)1 implies M(D)2. We find a sharp lower bound for M*(D) when D is locally raylike and a sharp upper bound for M*(D) when D is a reflection domain, a domain whose boundary admits a quasiconformal reflection. Based on these estimates, we are able to obtain the exact values of M(D) and M*() and for wedges, cones and related domains. For example, M(D) and M*(D) are equal to 6 or 8 when D is a regular triangle in \IR\sp2 or a cube in \IR\sp3, respectively. We also study some properties of reflection domains. In particular, by using the results about the estimates of M(D) and M*(D) mentioned above, we are able to find the extremal reflections for some domains such as regular n-gons, wedges and circular cones. This provides a new way to attack the important and difficult problem of finding extremal quasiconformal mappings between domains.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105631/1/9135725.pdfDescription of 9135725.pdf : Restricted to UM users only
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