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  • Cited by 349
Publisher:
Cambridge University Press
Online publication date:
September 2009
Print publication year:
2004
Online ISBN:
9780511546600

Book description

Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

Reviews

‘This book is devoted to the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.‘

Source: Monatshefte für Mathematik

‘For all students in this filed Duren's book will be essential reading. it will also be the classic reference book in this area.‘

Source: Proceedings of the Edinburgh Mathematical Society

'Those who are sensible to the beauty of complex functions and Riemann surfaces will certainly enjoy reading this nicely written … book.'

Source: Mathematical Geology

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Contents

References
Abu-Muhanna, Y. and Lyzzaik, A., A geometric criterion for decomposition and multivalence, Math. Proc. Cambridge Philos. Soc. 103 (1988), 487–495
Abu-Muhanna, Y. and Lyzzaik, A., The boundary behaviour of harmonic univalent maps, Pacific J. Math. 141 (1990), 1–20
Abu-Muhanna, Y. and Schober, G., Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), 1489–1530
L. V. Ahlfors, Lectures on Quasiconformal Mappings (Van Nostrand, Princeton, N.J., 1966)
L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973)
L. V. Ahlfors, Complex Analysis (Third Edition, McGraw-Hill, New York, 1979)
Avci, Y. and Złotkiewicz, E., On harmonic univalent functions, Ann. Univ. Mariae Curie-Skłodowska 44 (1990), 1–7
J. L. M. Barbosa and A. G. Colares, Minimal Surfaces in R3, Lecture Notes in Math. No. 1195 (Springer-Verlag, Berlin, 1986)
Bers, L., Isolated singularities of minimal surfaces, Ann. of Math. 53 (1951), 364–386
Bers, L., Univalent solutions of linear elliptic systems, Comm. Pure Appl. Math. 6 (1953), 513–526
Bojarski, B. V., Homeomorphic solutions of a Beltrami system, Dokl. Akad. Nauk SSSR 102 (1955), 661–664 (Russian)
Bojarski, B. V., On solutions of a linear elliptic system of differential equations in the plane, Dokl. Akad. Nauk SSSR 102 (1955), 871–874 (Russian)
Bojarski, B. V. and Iwaniec, T., Quasiconformal mappings and non-linear elliptic equations in two variables I, II, Bull. Polish Acad. Sci. Math. 22 (1974), 473–478, 479–484
Bshouty, D., Hengartner, N., and Hengartner, W., A constructive method for starlike harmonic mappings, Numer. Math. 54 (1988), 167–178
D. Bshouty and W. Hengartner, Boundary correspondence of univalent harmonic mappings from the unit disc onto a Jordan domain, in Approximation by Solutions of Partial Differential Equations, Quadrature Formulas, and Related Topics, B. Fuglede et al., eds., NATO ASI Series C, Vol. 365 (Kluwer Academic Publishers, Dordrecht-Boston-London, 1992), pp. 51–60
Bshouty, D. and Hengartner, W., Univalent solutions of the Dirichlet problem for ring domains, Complex Variables Theory Appl. 21 (1993), 159–169
Bshouty, D. and Hengartner, W., Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Skłodowska 48 (1994), 12–42
Bshouty, D. and Hengartner, W., Boundary values versus dilatations of harmonic mappings, J. Analyse Math. 72 (1997), 141–164
Bshouty, D. and Hengartner, W., Exterior univalent harmonic mappings with finite Blaschke dilatations, Canad. J. Math. 51 (1999), 470–487
Bshouty, D., Hengartner, W., and Hossian, O., Harmonic typically real mappings, Math. Proc. Cambridge Philos. Soc. 119 (1996), 673–680
Bshouty, D., Hengartner, W., and Suez, T., The exact bound on the number of zeros of harmonic polynomials, J. Analyse Math. 67 (1995), 207–218
Chen, H., Gauthier, P. M., and Hengartner, W., Bloch constants for planar harmonic mappings, Proc. Amer. Math. Soc. 128 (2000), 3231–3240
Choquet, G., Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. 69 (1945), 156–165
Choquet, G., Sur les homéomorphies harmoniques d'un disque D sur D, Complex Variables Theory Appl. 24 (1993), 47–48
Chuaqui, M., Duren, P., and Osgood, B., The Schwarzian derivative for harmonic mappings, J. Analyse Math. 91 (2003), 329–351
Cima, J. A. and Livingston, A. E., Integral smoothness properties of some harmonic mappings, Complex Variables Theory Appl. 11 (1989), 95–110
Cima, J. A. and Livingston, A. E., Nonbasic harmonic maps onto convex wedges, Colloq. Math. 66 (1993), 9–22
Clunie, J. and Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I 9 (1984), 3–25
Colonna, F., The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), 829–840
Vries, H. L., A remark concerning a lemma of Heinz on harmonic mappings, J. Math. Mech. 11 (1962), 469–471
Vries, H. L., Über Koeffizientenprobleme bei Eilinien und über die Heinzsche Konstante, Math. Z. 112 (1969), 101–106
U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal Surfaces I (Springer-Verlag, Berlin-Heidelberg-New York, 1991)
Dorff, M. J., Some harmonic n-slit mappings, Proc. Amer. Math. Soc. 126 (1998), 569–576
Dorff, M. J., Convolutions of planar harmonic convex mappings, Complex Variables Theory Appl. 45 (2001), 263–271
Dorff, M. J., Minimal graphs in ℝ3 over convex domains, Proc. Amer. Math. Soc. 132 (2004), 491–498
Dorff, M. J. and Suffridge, T. J., The inner mapping radius of harmonic mappings of the unit disk, Complex Variables Theory Appl. 33 (1997), 97–103
Dorff, M. and Szynal, J., Harmonic shears of elliptic integrals, Rocky Mountain J. Math., to appear
Driver, K. and Duren, P., Harmonic shears of regular polygons by hypergeometric functions, J. Math. Anal. Appl. 239 (1999), 72–84
P. L. Duren, Theory of HpSpaces (Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, N.Y., 2000)
P. L. Duren, Univalent Functions (Springer-Verlag, New York, 1983)
P. L. Duren, A survey of harmonic mappings in the plane, in Texas Tech University, Mathematics Series, Visiting Scholars' Lectures 1990–1992, vol. 18 (1992), pp. 1–15
Duren, P. and Hengartner, W., A decomposition theorem for planar harmonic mappings, Proc. Amer. Math. Soc. 124 (1996), 1191–1195
Duren, P. and Hengartner, W., Harmonic mappings of multiply connected domains, Pacific J. Math. 180 (1997), 201–220
Duren, P., Hengartner, W., and Laugesen, R. S., The argument principle for harmonic functions, Amer. Math. Monthly 103 (1996), 411–415
Duren, P. and Khavinson, D., Boundary correspondence and dilatation of harmonic mappings, Complex Variables Theory Appl. 33 (1997), 105–111
Duren, P. and Schober, G., A variational method for harmonic mappings onto convex regions, Complex Variables Theory Appl. 9 (1987), 153–168
Duren, P. and Schober, G., Linear extremal problems for harmonic mappings of the disk, Proc. Amer. Math. Soc. 106 (1989), 967–973
Duren, P. and Thygerson, W. R., Harmonic mappings related to Scherk's saddle-tower minimal surfaces, Rocky Mountain J. Math. 30 (2000), 555–564
Finn, R. and Osserman, R., On the Gauss curvature of non-parametric minimal surfaces, J. Analyse Math. 12 (1964), 351–364
FitzGerald, C. and Pommerenke, Ch., The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), 683–690
W. H. J. Fuchs, Topics in the Theory of Functions of One Complex Variable (Van Nostrand, Princeton, N.J., 1967)
Gleason, S. and Wolff, T. H., Lewy's harmonic gradient maps in higher dimensions, Comm. Partial Differential Equations 16 (1991), 1925–1968
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (Moscow, 1952; German transl., Deutscher Verlag, Berlin, 1957; Second Edition, Izdat. “Nauka”, Moscow, 1966; English transl., American Mathematical Society, Providence, R.I., 1969)
Goodloe, M. R., Hadamard products of convex harmonic mappings, Complex Variables Theory Appl. 47 (2002), 81–92
Goodman, A. W. and Saff, E. B., On univalent functions convex in one direction, Proc. Amer. Math. Soc. 73 (1979), 183–187
P. Greiner, Boundary properties of planar harmonic mappings, Ph. D. Thesis, University of Michigan, 1995
Greiner, P., Geometric properties of harmonic shears, Computational Methods and Function Theory, to appear
Grigoryan, A. and Nowak, M., Integral means of harmonic mappings, Ann. Univ. Mariae Curie-Skł, odowska 52 (1998), 25–34
Grigoryan, A. and Nowak, M., Estimates of integral means of harmonic mappings, Complex Variables Theory Appl. 42 (2000), 151–161
Grigoryan, A. and Szapiel, W., Two-slit harmonic mappings, Ann. Univ. Mariae Curie-Skł, odowska 49 (1995), 59–84
Hall, R. R., On a conjecture of Shapiro about trigonometric series, J. London Math. Soc. 25 (1982), 407–415
Hall, R. R., On an inequality of E. Heinz, J. Analyse Math. 42 (1982/83), 185–198
Hall, R. R., A class of isoperimetric inequalities, J. Analyse Math. 45 (1985), 169–180
Hall, R. R., The Gaussian curvature of minimal surfaces and Heinz' constant, J. Reine Angew. Math. 502 (1998), 19–28
W. K. Hayman, Multivalent Functions (Cambridge University Press, London, 1958)
Heinz, E., Über die Lösungen der Minimalflächengleichung, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1952), 51–56
Heinz, E., On one-to-one harmonic mappings, Pacific J. Math. 9 (1959), 101–105
Hengartner, W. and Schober, G., On schlicht mappings to domains convex in one direction, Comm. Math. Helv. 45 (1970), 303–314
Hengartner, W. and Schober, G., A remark on level curves for domains convex in one direction, Appl. Analysis 3 (1973), 101–106
Hengartner, W. and Schober, G., Univalent harmonic mappings onto parallel slit domains, Michigan Math. J. 32 (1985), 131–134
Hengartner, W. and Schober, G., On the boundary behavior of orientation-preserving harmonic mappings, Complex Variables Theory Appl. 5 (1986), 197–208
Hengartner, W. and Schober, G., Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473–483
Hengartner, W. and Schober, G., Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1–31
W. Hengartner and G. Schober, Curvature estimates for some minimal surfaces, in Complex Analysis: Articles Dedicated to Albert Pfluger on the Occasion of his 80th Birthday, J. Hersch and A. Huber, editors (Birkhäuser Verlag, Basel, 1988), pp. 87–100
Hengartner, W. and Schober, G., Univalent harmonic exterior and ring mappings, J. Math. Anal. Appl. 156 (1991), 154–171
Hengartner, W. and Szynal, J., Univalent harmonic ring mappings vanishing on the interior boundary, Canad. J. Math. 44 (1992), 308–323
Hersch, J. and Pfluger, A., Généralisation du lemme de Schwarz et du principe de la mesure harmonique pour les fonctions pseudo-analytiques, C. R. Acad. Sci. Paris 234 (1952), 43–45
Hildebrandt, S. and Sauvigny, F., Embeddedness and uniqueness of minimal surfaces solving a partially free boundary value problem, J. Reine Angew. Math. 422 (1991), 69–89
Hoffman, D., The computer-aided discovery of new embedded minimal surfaces, Math. Intelligencer 9 (1987), 8–21
Hopf, E., On an inequality for minimal surfaces z = z(x, y), J. Rational Mech. Anal. 2 (1953), 519–522; 801–802
Jahangiri, J. M., Morgan, C., and Suffridge, T. J., Construction of close-to-convex harmonic polynomials, Complex Variables Theory Appl. 45 (2001), 319–326
S. H. Jun, Harmonic mappings and applications to minimal surfaces, Ph. D. Thesis, Indiana University, 1989
Jun, S. H., Curvature estimates for minimal surfaces, Proc. Amer. Math. Soc. 114 (1992), 527–533
Jun, S. H., Univalent harmonic mappings on Δ = }z: |z| > 1{, Proc. Amer. Math. Soc. 119 (1993), 109–114
Jun, S. H., Planar harmonic mappings and curvature estimates, J. Korean Math. Soc. 32 (1995), 803–814
Khavinson, D. and Swiatek, G., On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), 409–414
Kneser, H., Lösung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 123–124
Krzyż, J., and Nowak, M., Harmonic automorphisms of the unit disk, J. Comput. Appl. Math. 105 (1999), 337–346
Laugesen, R. S., Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl. 28 (1996), 357–369
Laugesen, R. S., Planar harmonic maps with inner and Blaschke dilatations, J. London Math. Soc. 56 (1997), 37–48
O. Lehto, Univalent Functions and Teichmüller Spaces (Springer-Verlag, New York, 1987)
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane (Second Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1973)
Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692
Lewy, H., On the non-vanishing of the jacobian of a homeomorphism by harmonic gradients, Ann. of Math. 88 (1968), 518–529
Liu, H. and Liao, G., A note on harmonic maps, Appl. Math. Lett. 9 (1996), no. 4, 95–97
Livingston, A. E., Univalent harmonic mappings, Ann. Polon. Math. 57 (1992), 57–70
Livingston, A. E., Univalent harmonic mappings II, Ann. Polon. Math. 67 (1997), 131–145
Lyzzaik, A., On the valence of some classes of harmonic maps, Math. Proc. Cambridge Philos. Soc. 110 (1991), 313–325
Lyzzaik, A., Local properties of light harmonic mappings, Canad. J. Math. 44 (1992), 135–153
Lyzzaik, A., The geometry of some classes of folding polynomials, Complex Variables Theory Appl. 20 (1992), 145–155
Lyzzaik, A., Univalence criteria for harmonic mappings in multiply-connected domains, J. London Math. Soc. 58 (1998), 163–171
Lyzzaik, A., A note on the valency of harmonic maps, J. Math. Anal. Appl. 218 (1998), 611–620
Lyzzaik, A., The modulus of the image annuli under univalent harmonic mappings and a conjecture of J. C. C. Nitsche, J. London Math. Soc. 64 (2001), 369–384
Martio, O., On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I, (1968), 3–10
Mateljević, M. and Pavlović, M., Multipliers of Hp and BMOA, Pacific J. Math. 146 (1990), 71–84
Melas, A. D., An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc. 117 (1993), 857–859
Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551
Z. Nehari Conformal Mapping (McGraw-Hill, New York, 1952; reprinted by Dover Publications, New York, 1975)
Neumann, G., Valence of complex-valued planar harmonic functions, to appear
Nitsche, J. C. C., Über eine mit der Minimalflächengleichung zusammenhängende analytische Funktion und den Bernsteinschen Satz, Archiv der Math. (Basel) 7 (1956), 417–419
Nitsche, J. C. C., On harmonic mappings, Proc. Amer. Math. Soc. 9 (1958), 268–271
Nitsche, J. C. C., On an estimate for the curvature of minimal surfaces z = z(x, y), J. Math. Mech. 7 (1958), 767–769
Nitsche, J. C. C., On the constant of E. Heinz, Rend. Circ. Mat. Palermo 8 (1959), 178–181
Nitsche, J. C. C., On the module of doubly-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), 781–782
Nitsche, J. C. C., Zum Heinzschen Lemma über harmonische Abbildungen, Arch. Math. (Basel) 14 (1963), 407–410
J. C. C. Nitsche, Lectures on Minimal Surfaces, Vol. I (Cambridge University Press, Cambridge, 1989)
Nowak, M., Integral means of univalent harmonic maps, Ann. Univ. Mariae Curie-Skł, odowska 50 (1996), 155–162
Osserman, R., On the Gauss curvature of minimal surfaces, Trans. Amer. Math. Soc. 96 (1960), 115–128
Osserman, R., Global properties of classical minimal surfaces, Duke Math. J. 32 (1965), 565–573
R. Osserman, A Survey of Minimal Surfaces (Second Edition, Dover Publications, Mineola, N.Y., 1986)
R. Osserman, Minimal surfaces in R3, in Global Differential Geometry, S. S. Chern, editor, Mathematical Association of America Studies in Mathematics Vol. 27 (1989), pp. 73–98
G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, N.J., 1951)
Ch. Pommerenke, Univalent Functions (Vandenhoeck & Ruprecht, Göttingen, 1975)
Radó, T., Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 49
Radó, T., Über den analytischen Charakter der Minimalflächen, Math. Z. 24 (1926), 321–327
Radó, T., Zu einem Satze von S. Bernstein über Minimalflächen im Grossen, Math. Z. 26 (1927), 559–565
Reich, E., The composition of harmonic mappings, Ann. Acad. Sci. Fenn. Ser. A.I 12 (1987), 47–53
Reich, E., Local decomposition of harmonic mappings, Complex Variables Theory Appl. 9 (1987), 263–269
H. Renelt, Quasikonforme Abbildungen und elliptische Systeme erster Ordnung in der Ebene (B. G. Teubner, Leipzig, 1982); English edition: Elliptic Systems and Quasiconformal Mappings (John Wiley & Sons, New York, 1988)
Robertson, M. S., Analytic functions star-like in one direction, Amer. J. Math. 58 (1936), 465–472
Royster, W. C. and Ziegler, M., Univalent functions convex in one direction, Publ. Math. Debrecen 23 (1976), 339–345
W. Rudin, Principles of Mathematical Analysis (Third Edition, McGraw-Hill, New York, 1976)
Ruscheweyh, St. and Salinas, L., On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. A.I 14 (1989), 63–73
Schaubroeck, L. E., Subordination of planar harmonic functions, Complex Variables Theory Appl. 41 (2000), 163–178
Schaubroeck, L. E., Growth, distortion and coefficient bounds for plane harmonic mappings convex in one direction, Rocky Mountain J. Math. 31 (2001), 625–639
G. Schober, Planar harmonic mappings, in Computational Methods and Function Theory, Lecture Notes in Math. No. 1435 (Springer-Verlag, Berlin-Heidelberg, 1990), pp. 171–176
J. T. Schwartz, Nonlinear Functional Analysis (Gordon and Breach, New York, 1969)
H. S. Shapiro, Research problems in complex analysis (edited by J. M. Anderson, K. F. Barth, and D. A. Brannan), Bull. London Math. Soc. 9 (1977), 129–162; Problem No. 7.26, p. 146
Sheil-Small, T., On the Fourier series of a finitely described convex curve and a conjecture of H. S. Shapiro, Math. Proc. Cambridge Philos. Soc. 98 (1985), 513–527
Sheil, T.-Small, On the Fourier series of a step function, Michigan Math. J. 36 (1989), 459–475
Sheil, T.-Small, Constants for planar harmonic mappings, J. London Math. Soc. 42 (1990), 237–248
J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969)
D. J. Struik, Lectures on Classical Differential Geometry (Second Edition, Addison-Wesley, Cambridge, Mass., 1961; reprinted by Dover Publications, Mineola, N.Y., 1988)
Suffridge, T. J., Harmonic univalent polynomials, Complex Variables Theory Appl. 35 (1998), 93–107
Suffridge, T. J. and Thompson, J. W., Local behavior of harmonic mappings, Complex Variables Theory Appl. 41 (2000), 63–80
Szulkin, A., An example concerning the topological character of the zero-set of a harmonic function, Math. Scand. 43 (1978), 60–62
Ullman, J. L. and Titus, C. J., An integral inequality with applications to harmonic mappings, Michigan Math. J. 10 (1963), 181–192
N. Vekua, Generalized Analytic Functions (Pergamon Press, London, 1962)
A. Weitsman, Harmonic mappings whose dilatations are singular inner functions, preprint (1996)
Weitsman, A., On the dilatation of univalent planar harmonic mappings, Proc. Amer. Math. Soc. 126 (1998), 447–452
Weitsman, A., On the Fourier coefficients of homeomorphisms of the circle, Math. Res. Lett. 5 (1998), 383–390
Weitsman, A., On univalent harmonic mappings and minimal surfaces, Pacific J. Math. 192 (2000), 191–200
Weitsman, A., Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche, Israel J. Math. 124 (2001), 327–331
Weitsman, A., On the Poisson integral of step functions and minimal surfaces, Canad. Math. Bull. 45 (2002), 154–160
W. L. Wendland, Elliptic Systems in the Plane (Pitman, London, 1979)
Wilmshurst, A., The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), 2077–2081
Wood, J. C., Lewy's theorem fails in higher dimensions, Math. Scand. 69 (1991), 166

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