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Keywords:
Gaussian curvature; holomorphic function
Gaussian curvature; holomorphic function
Summary:
It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a graph with radially symmetric nonpositive Gaussian curvature in a certain form, there are (up to) four families of harmonic functions whose graphs have this curvature. Moreover, the graphs obtained from these functions are not isometric in general.
References:
[1] Conway, J. B.: Functions of One Complex Variable I. Second edition. Springer, New York, 1973. MR 0447532
[2] Millman, R. S., Parker, G. D.: Elements of Differential Geometry. Prentice-Hall, New Jersey, 1977. MR 0442832
[3] Pressley, A.: Elementary Differential Geometry. Springer, London, 2001. MR 1800436 | Zbl 0959.53001
[4] Weissteins’, E.: Mathworld. Wolfram Research, Inc. CRC Press LLC, http://mathworld.wolfram.com, 1999.
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