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BOUNDS FOR JACOBIAN OF HARMONIC INJECTIVE

MAPPINGS IN N-DIMENSIONAL SPACE

VLADIMIR BOZIN, MIODRAG MATELJEVIC

Research partially supported by MNTRS, Serbia, Grant No. 174 032

1. Introduction

In his seminal paper, Olli Martio [11] observed that every quasiconformal har-monic mapping of the unit planar disk onto itself is co-Lipschitz. Later, the subjectof quasiconformal harmonic mappings was intensively studied by the participants ofthe Belgrade Analysis Seminar, see for example [10, 14, 15, 17].

If you want you can use my letter; I wrote it in honest thoughts, what I felt of the unfortunate and unfair situation you are facing. So in that sense it was defending the truth.

Internationally your result is by far the best known in the field of HQC-maps, and therefore it was a big surprise when I noticed that the works of Prof. Mateljevic systematically try to hide the strong influence of your paper. This felt very bad to me, I must confess. Whether you call it "plagiarism" or "efforts of forgery" or similar is a matter of taste.

I do not know if it brings to you any consolation, but internationally your above work is still considered as the landmark in HQC-maps, much more than any other paper in the field. Therefore its role cannot be changed internationally. The more so, for an outsider it gives a very bad impression if the role of your paper is purposely tried to hidden - again for the outsiders that kind of behaviour has the opposite effect of working against such authors

Confidentally: We have not cited Pavlović's paper "Boundary correspondence of HQC-homeomorphism of the disk"

THE LOWER BOUND FOR JACOBIAN 7

from Theorem 2.1, we get a uniform bound from below for norm of the derivativeof h, and hence conclude that map is co-Lipschitz. �

Now, the special case of interest we get by applying Theorems 3.3 and 3.2.

Theorem 4.4. Suppose h is a harmonic K-quasiconformal mapping from the unitball Bn onto a bounded convex domain D = h(Bn), that is gradient of a harmonicfunction. Then h is co-Lipschitz on Bn.

Theorem 4.5. Suppose h is a harmonic K-quasiconformal mapping from the unitball Bn onto a bounded convex domain D = h(Bn), with K < 3n−1. Then h isco-Lipschitz on B

n.

References

[1] K. Astala, V. Manojlovic, Pavlovic’s theorem in space, arXiv:1410.7575 [math.CV][2] P. Duren, Harmonic mappings in the plane, Cambridge Univ. Press, 2004.[3] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order

Springer-Verlag Berlin Heidelberg New York Tokyo, 2001.[4] T. Iwaniec and J. Onninen, Rado-Kneser-Choquet theorem, Bull. London Math. Soc. 46(2014) 1283-1291

[5] J. Jost, Harmonic Maps Between surfaces, Springer-Verlag Berlin Heidelberg New YorkTokyo, 1984.

[6] J. Jost, Two-dimensional Geometric Variational Problems, John Wiley & Sons, 1991.[7] D. Kalaj, M. S. Mateljevic, Harmonic quasiconformal self-mappings and Mobius trans-

formations of the unit ball. Pacific journal of mathematics. ISSN 0030-8730. 247 : 2 (2010),389-406.

[8] D. Kalaj, A priori estimate of gradient of a solution to certain differential inequality and

quasiregular mappings, Journal d’Analyse Mathematique, (2013), Volume 119, Issue 1, pp. 63-88.[9] V. Markovic Harmonic maps between 3-dimensional hyperbolic spaces. Inventiones Mathe-maticae, DOI 10.1007/s00222-014-0536-x, http://arxiv.org/pdf/1308.1710.pdf.

[10] V. Markovic, M. Mateljevic, A new version of the main inequality and the uniqueness of

harmonic maps, Journal d’Analyse Mathematique, (1999), Volume 79, Issue 1, pp. 315-334.[11] O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn., A 1, 425, 3-10(1968)

[12] M. Mateljevic, Estimates for the modulus of the derivatives of harmonic univalent map-

pings, Proceedings of International Conference on Complex Analysis and Related Topics (IXth

Romanian-Finnish Seminar, 2001), Rev Roum Math Pures Appliq (Romanian Journal of Pureand Applied mathematics) 47 (2002) 5-6, 709 -711.

[13] M. Mateljevic, Distortion of harmonic functions and harmonic quasiconformal quasi-

isometry, Revue Roum. Math. Pures Appl. Vol. 51 (2006) 5–6, 711–722.[14] M. Mateljevic, Quasiconformality of harmonic mappings between Jordan domains 2, Filo-mat 26:3 (2012) 479-509.

[15] M. Mateljevic, Topics in Conformal, Quasiconformal and Harmonic maps, Zavod zaudzbenike, Beograd 2012.

[16] M. Mateljevic, Distortion of quasiregular mappings and equivalent norms on Lipschitz-type

spaces, Abstract and Applied Analysis Volume 2014 (2014), Article ID 895074, 20[17] M. Mateljevic, The lower bound for the modulus of the derivatives and Jacobian of har-

monic univalent mappings http://arxiv.org/pdf/1501.03197v1.pdf, [math.CV] 13 Jan 2015[18] L. Tam, T. Wan, On quasiconformal harmonic maps, Pacific J Math, V 182 (1998), 359-383[19] J. Vaisala, Lectures on n-Dimensional Quasiconformal Mappings, Springer-Verlag, 1971.[20] M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Math. V1319, Springer-Verlag, 1988.

[21] J. C. Wood, Lewy’s theorem fails in higher dimensions, Math. Scand. 69 (1991), 166.

Faculty of mathematics, Univ. of Belgrade, Studentski Trg 16, Belgrade, Serbia

E-mail address: miodrag@matf.bg.ac.rs