Abstract
We introduce the comparison and the tangency principles that will be ones of the main tools employed in this book. They are obtained when writing locally a surface as the graph of a function z=u(x,y). Then the constancy of the mean curvature on the surface implies that u satisfies a second order partial differential equation of elliptic type. The ellipticity of the equation allows the use of the classical Hopf maximum principle. With the aid of these techniques, the problem studied in this chapter is what type of consequences can be deduced when two surfaces with the same constant mean curvature are tangent at a common point. We will obtain a first set of results on compact cmc surfaces with planar boundary giving conditions that ensure that the surface lies on one side of the boundary plane. Comparing the surface with spheres and cylinders with the same mean curvature, we obtain characterizations of a sphere if the surface is included in the closure of an Euclidean ball or a cylinder whose radii are related with the value of the mean curvature of the surface.
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References
Almgren, F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84, 277–292 (1966)
Bernstein, S.N.: Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique. Commun. Soc. Math. Kharkov 15, 38–45 (1915–1917)
Finn, R.: Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature. J. Anal. Math. 14, 139–160 (1965)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). Reprint of the 1998 edition
Greenberg, M., Harper, J.: Algebraic Topology: A First Course. Benjamin-Cummings, Reading (1981)
Hopf, E.: Elementare Bemerkumgen Über Die Lösungen Partieller Differential Gleichungen. S. B. Preuss. Akad. Phys. Math. Kl., 147–152 (1927)
Hopf, E.: Über Die Lösungen del Minimalflächengleichung. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl., 2B II, 51–56 (1952)
Hopf, H.: Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000. Springer, Berlin (1983)
Koiso, M.: Symmetry of hypersurfaces of constant mean curvature with symmetric boundary. Math. Z. 191, 567–574 (1986)
Koutroufiotis, D.: Elementary geometric applications of a maximum principle for nonlinear elliptic operators. Arch. Math. 24, 97–99 (1973)
Osserman, R.: A Survey of Minimal Surfaces. Dover, New York (1986)
Protter, M.H., Weinberger, H.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)
Serrin, J.: On surfaces of constant mean curvature which span a given space curve. Math. Z. 112, 77–88 (1969)
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López, R. (2013). The Comparison and Tangency Principles. In: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39626-7_3
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DOI: https://doi.org/10.1007/978-3-642-39626-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39625-0
Online ISBN: 978-3-642-39626-7
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