August 9, 1995
We survey what is known about minimal surfaces in $\bold R^3 $ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples.
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The study of embedded minimal surfaces in $\RR^3$ is a classical problem, dating to the mid 1700's, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and Courant, and taking the opportunity to focus on results that have not been highlighted elsewhere.
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An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case.
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We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegenerate. Finally, using the same techniques, we are able to produce properly embedded minimal surfaces with infinitely many ends. Each annular end has finite to...
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July 10, 2014
Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. This paper is motivated by two outstanding features of such surfaces:...
July 22, 2019
We extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular, we extend the seminal works of L. Jorge and W. Meeks and R. Schoen. Specifically, we extend the Jorge-Meeks formula relating the total curvature and the topology of the surface and we use it to classify planes as the only elliptic special Weingarten surfaces whose total curvature is...
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In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}...
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In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (1995). As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in Kapouleas (1990) and we produce a very large class of new embedded examples o...
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A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known classification results for properly embedded minimal surfaces with genus zero in $\mathbb{R}^3$ or quotients of $\mathbb{R}^3$ by one or two independent translations. This does not intend to be an exhaustive review of the tools or proofs in the field,...