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Hyperbolic Weingarten surfaces

Published online by Cambridge University Press:  24 October 2008

B. Van-Brunt  and
K. Grant
B. Van-Brunt
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand
K. Grant
Affiliation:
Department of Mathematics, Massey University, Palmerston North, New Zealand
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Abstract

Weingarten surfaces which can be represented locally as solutions to second order hyperbolic partial differential equations are examined in this paper. In particular, the geometry of the families of curves corresponding to characteristics on these surfaces is investigated and the relationships of these curves with other curves on the surface such as asymptotic lines and lines of curvature are explored. It is shown that singularities in the lines of curvature, i.e. umbilic points, correspond to singularities in the families of characteristics, and that lines of curvature are non-characteristic curves. If there is a linear relation between the Gaussian and mean curvatures and real characteristics exist, then the characteristics form a Tchebychef net on the corresponding Weingarten surface.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 489 - 504
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Caffarelli, L., Nirenberg, L. and Spruck, J.. Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces. Comm. Pure Appl. XLI (1988), 47–70.CrossRefGoogle Scholar
[2]Caffarelli, L., Nirenberg, L. and Spruck, J.. Nonlinear second-order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces. Current Topics in Partial Differential Equations, Kinokunize Co. Tokyo (1986), 1–26.Google Scholar
[3]Chern, S. S.. Some new characterizations of the Euclidean sphere. Duke Math. J. 12 (1945), 279–290.CrossRefGoogle Scholar
[4]Chern, S. S.. On special Weingarten surfaces. Proc. Amer. Math. Soc. 6 (1955), 783–786.Google Scholar
[5]Coddington, E. and Levinson, N.. Theory of Ordinary Differential Equations (McGraw-Hill, 1955).Google Scholar
[6]Darboux, G.. Leçons sur la Théorie Général des Surfaces et les Applications Géométriques du calcul Infinitésimal, vol. III (Gauthier-Villars, 1894).Google Scholar
[7]Eisenhart, L.. A Treatise on the Differential Geometry of Curves and Surfaces (Ginn, 1909).Google Scholar
[8]Forsyth, A.. Lectures on the Differential Geometry of Curves and Surfaces (Cambridge University Press, 1912).Google Scholar
[9]Hartman, P. and Wintner, A.. Umbilical Points and W-Surfaces. Amer. J. Math. 76 (1954), 502–508.CrossRefGoogle Scholar
[10]Hopf, H.. Differential Geometry in the Large, 2nd ed. (Springer-Verlag, 1989).CrossRefGoogle Scholar
[11]Hopf, H.. Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4 (1951), 232–249.CrossRefGoogle Scholar
[12]Korevaar, N. J.. A priori interior gradient bounds for solutions to elliptic Weingarten equations. Ann. Inst. H. Poincare Anal. Non Linéaire 4, no. 5 (1987), 405–421.CrossRefGoogle Scholar
[13]Milnor, T. K.. Abstract Weingarten surfaces. J. Differential Geom. 15, no. 3 (1980), 365–380.CrossRefGoogle Scholar
[14]Palais, R. and Terng, C.. Critical point theory and submanifold geometry, Lecture Notes in Mathematics No. 1353 (Springer-Verlag, 1988).CrossRefGoogle Scholar
[15]Ŝvec, A.. On equiaffine Weingarten surfaces. Czechoslovak Math. J. 37 (122), no. 4 (1987), 567–572.CrossRefGoogle Scholar
[16]Svec, A.. Global differential geometry of surfaces (D. Reidel, 1981).CrossRefGoogle Scholar
[17]Voss, K.. Uber geschlossene Weingartensche Flachen. Math. Ann. 138 (1959), 42–54.CrossRefGoogle Scholar