Minimal Surfaces in R3 Francisco Martín Program References

In this course we will review some topics on the theory of minimal surfaces in three dimensional Euclidean space.

The starting point of the study of minimal surfaces was the problem of determining a graph, z = f(x,y), over an open set U in R², with the least possible area among all surfaces that assume given values on the boundary of U. Using the Euler-Lagrange equations, one can easily see that f must satisfy the following quasilinear second order partial differential equation:

Mathematicians soon realized that here was not only a problem of extraordinary difficulty, but also of unlimited possibilities.

This initial approach is analytic, but we can supply a geometric interpretation of the minimal graph equation (1): the mean curvature of the surface, H, vanishes. On this premise it has become customary to use the term minimal surface for any surface satisfying H = 0, notwithstanding the fact that such surfaces often do not provide a minimum for the area.

In 1847, the Belgian physicist J. Plateau observed that minimal surfaces could be physically produced by dipping a curved wire frame into a solution of soapy water and glycerin. So, the problem of determining a minimal surface with fixed topology and bounded by a prescribed Jordan curve is now usually called Plateau's problem. Plateau did not have the mathematical skills to investigate the problem theoretically but Weierstrass, Riemann and Schwarz worked on the problem which was finally solved by Douglas and Radó.

To end this small list of characterizations, we would like to point out that the Gauss map of a minimal surface is conformal, and that this property characterizes minimal surfaces, besides the round sphere. This fact has a crucial importance in the study of minimal surfaces in R³. One of the fundamental problems in classical theory of minimal surfaces has been to obtain Liouville type results for the Gauss map of complete minimal surfaces.

However, minimal surfaces are not only interesting from a mathematical point of view, they are also important in Physics, Chemistry, Biology and Engineering. For instance, minimal surfaces have been used to model liquid crystals that arise in oil-water-surfactant microemulsions and to model dividing surfaces in certain microemulsions of block copolymers.

This course will be devoted to some aspects of the theory of minimal surfaces. As we will see, equation (1) has a strong influence on the topology, conformal structure and other geometrical properties of a minimal surface.

A minimal surface can, at least locally, be seen as a solution of (1). This fact will allow us to deduce a Maximum Principle for minimal surfaces which has been a fundamental tool in obtaining a large number of results in this field. We emphasize the following topics:
 

• The convex hull property (C.H.P.). We say that a surface satisfies the C.H.P. if every connected, compact region is contained in the convex hull of its boundary. We will proof a theorem by R. Osserman (J. Differential Geom., 1979) that asserts:


A surface M in R³ satisfies the C.H.P. if, and only if, the Gauss curvature of M is non positive.

Furthermore, we will get some recent generalizations of this result.
 

• The strong half space theorem. One of the more recent and nicer applications of the maximum principle is the following result by Hoffman and Meeks (Invent. mathem., 1990):


A connected, proper, non-planar, complete minimal surface in R³ is not contained in a half space.
 

• Alexandrov's theorem. In 1956, Alexandrov proved that the sphere is the unique embedded, compact surface in R³ with constant mean curvature. This result was conjectured by Hilbert and, historically, it was the first application of the Maximum Principle in Geometry.

1. Preliminaries.
• Basic results on surface theory.
• Basic results on P.D.E.
2. Applications of the Maximum Principle to the geometry of minimal surfaces.
• The Convex Hull Property.
• The strong half space theorem.
• Some about Plateau's problem.
• Alexandrov's theorem.
• The isoperimetric inequality on minimal surfaces.
 

1. U. Dierkes et al., Minimal surfaces, Vol. I and II, Grundl. der mathem. Wiss. 295-6. Springer-Verlag, Heidelberg, 1992.
2. D. Hoffman & H. Karcher, Complete embedded minimal surfaces of finite total curvature. Encyclopaedia of Math. Sci., Vol. 90, R. Osserman (ed.), pp. 5-93, Springer-Verlag, Berlin, 1997.
3. F. J. López & F. Martín, Complete minimal surfaces in R³. Publicacions Matemàtiques (U. A. B.), Vol. 43, (2), (1999) 341-449. This paper is available via http://www.ugr.es/local/fmartin/complete_minimal_surfaces_in_r3.htm.
4. W. H. Meeks III, Lectures on Plateau's problem. Scola de Geometria Diferencial, Universidade Federal do Ceará (Brazil), 1978.
5. J.C.C. Nitsche, Lectures on minimal surfaces, Vol. I. Cambridge Univ. Press, Cambridge, 1989.
6. R. Osserman, A survey of minimal surfaces. Dover, New York, 1986.

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