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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Minimal graphs in $\mathbb{R}^3$ over convex domains


Author: Michael Dorff
Journal: Proc. Amer. Math. Soc. 132 (2004), 491-498
MSC (2000): Primary 49Q05, 53A10, 30C45
Published electronically: June 18, 2003
MathSciNet review: 2022374
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Abstract: Krust established that all conjugate and associate surfaces of a minimal graph over a convex domain are also graphs. Using a convolution theorem from the theory of harmonic univalent mappings, we generalize Krust's theorem to include the family of convolution surfaces which are generated by taking the Hadamard product or convolution of mappings. Since this convolution involves convex univalent analytic mappings, this family of convolution surfaces is much larger than just the family of associated surfaces. Also, this generalization guarantees that all the resulting surfaces are over close-to-convex domains. In particular, all the associate surfaces and certain Goursat transformation surfaces of a minimal graph over a convex domain are over close-to-convex domains.


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Additional Information

Michael Dorff
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602-6539
Email: mdorff@math.byu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07109-0
PII: S 0002-9939(03)07109-0
Received by editor(s): December 19, 2000
Received by editor(s) in revised form: October 14, 2002
Published electronically: June 18, 2003
Additional Notes: This work was supported in part by a grant from the University of Missouri Research Board
The author thanks the referee for his suggestions
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society