Conformal properties in classical minimal surface theory

Abstract

Abstract. This is a survey of recent developments in the classical theory of minimal surfaces in R 3 with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover the maximum principle at infinity for properly immersed minimal surfaces in R 3 and some new results on harmonic functions as they relate to the classical theory. We define and demonstrate the usefulness of universal superharmonic functions. We present the compactness and regularity theory of Colding and Minicozzi for limits of sequences of simply connected minimal surfaces and its application by Meeks and Rosenberg in their proof of the uniqueness of the helicoid. Finally, we discuss some recent deep results on the topology and on the index of the stability operator of properly embedded minimal surfaces and give an application of the classical Shiffman Jacobi function to the classification of minimal surfaces of genus zero.

Minimal surfaces with the area growth of two planes: The case of infinite symmetry

“…Consider the surfaces M n to lie in B( 0, r n ) ⊂ E 3 . Since the M n have maximal absolute curvature 1 at the origin and in balls of radius r ≤ r n have area at most 2πr 2 , standard results (see, for example, [8]) imply that a subsequence of these surfaces converges on compact subsets of E 3 to a properly embedded minimal surface M in E 3 with absolute curvature at most 1 and with absolute curvature 1 at the origin. The surface M is connected by the strong half-space theorem [1].…”

Eigenvalues of elliptic operators and geometric applications

“…Before we do so, let us briefly recall some results about the structure of minimal surfaces in R 3 . We refer a reader to the surveys by Hoffman and Karcher [30] and by Meeks and Perez [43] for more details.…”

“…By definition, the stability index ind (X) of the minimal surface X is Neg (L). It is well known that if ind (X) < ∞ then the total curvature 4 For a detailed account of minimal surfaces see the surveys [13] and [43] in the same volume.…”

“…Let d 0 be the geodesic distance on X with respect to the induced metric. Denote by d the extrinsic distance on X, that is the restriction to X of the Euclidean distance in Since X is complete and K total (X) < ∞, the immersion of X into R 3 is proper, that is the intersection of any compact set in R 3 with X is compact in the topology of X (see [43,Section 2.3]). This immediately implies that d-balls in X are precompact, that is the hypothesis (b).…”

Parabolicity of maximal surfaces in Lorentzian product spaces

“…It is interesting to observe that the parabolicity of is equivalent to the existence of a proper non-negative superharmonic function on (see beginning of Sect. 3; for details see [10,11]). …”