Minimal graphs in over convex domains

Author:
Michael Dorff

Journal:
Proc. Amer. Math. Soc. **132** (2004), 491-498

MSC (2000):
Primary 49Q05, 53A10, 30C45

DOI:
https://doi.org/10.1090/S0002-9939-03-07109-0

Published electronically:
June 18, 2003

MathSciNet review:
2022374

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Berglund, J. and W. Rossman, Minimal surfaces with catenoid ends,*Pacific J. Math.***171**, no. 2, (1995), 353-371. MR**97a:53007****2.**Clunie, J. and T. Sheil-Small, Harmonic univalent functions,*Ann. Acad. Sci. Fenn. Ser. A.I Math.***9**(1984), 3-25. MR**85i:30014****3.**Dierkes, U., S. Hildebrandt, A. Küster, and O. Wohlrab,*Minimal Surfaces I*, Grundlehren der mathematischen Wissenschaften, vol. 295, Springer-Verlag, Berlin-Heidelberg-New York, 1991. MR**94c:49001a****4.**Duren, P., A survey of harmonic mappings in the plane,*Texas Tech. Univ., Math. Series, Visiting Scholars Lectures, 1990-1992***18**(1992), 1-15.**5.**Duren, P.*Harmonic Mappings in the Plane*, in preparation.**6.**Duren, P. and W. R. Thygerson. Harmonic mappings related to Scherk's saddle-tower minimal surface,*Rocky Mountain J. Math.***30**(2000), 555-564. MR**2001i:58019****7.**Karcher, H.,*Construction of minimal surfaces*, Lecture Notes no. 12, SFB256, Bonn, 1989.**8.**Karcher, H., Construction of higher genus embedded minimal surfaces in*Geometry and topology of submanifolds, III (Leeds, 1990)*, 174-191, World Sci. Publishing, River Edge, NJ, 1991. MR**96e:53012****9.**Karcher, H. and Polthier, K., Construction of triply periodic minimal surfaces,*Philos. Trans. Roy. Soc. London*A**354**(1996), 2077-2104. MR**97i:53008****10.**Nitsche, J. C. C.,*Lectures on Minimal Surfaces*, vol. 1 , Cambridge University Press, Cambridge, 1989. MR**90m:49031****11.**Osserman, R.,*A Survey of Minimal Surfaces*, Dover Publications, New York, 1986. MR**87j:53012****12.**Rossman, W., Minimal surfaces in with dihedral symmetry,*Tohoku Math. J. (2)***47**, no. 1, (1995), 31-54. MR**95m:53009****13.**Rossman, W., New examples of minimal surfaces, in*Geometry and global analysis*, Tohoku Univ., Sendai, 1993, pp. 369-377. MR**96f:53015****14.**Sheil-Small, T., Constants for planar harmonic mappings,*J. London Math. Soc. 2***42**(1990), 237-248. MR**91k:30052****15.**Wei, Fu Sheng, Some existence and uniqueness theorems for doubly periodic minimal surfaces,*Invent. Math.***109**, no. 1, (1992), 113-136. MR**93c:53004**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
49Q05,
53A10,
30C45

Retrieve articles in all journals with MSC (2000): 49Q05, 53A10, 30C45

Additional Information

**Michael Dorff**

Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602-6539

Email:
mdorff@math.byu.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07109-0

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