English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It has recently been shown by Fischer-Colbrie and Schoen, and independently by Do Carmo and Peng, that if F is a complete stable minimal surface in R then F is a plane. Fischer-Colbrie and Schoen [FC-S] have obtained a similar result for 3-manifolds of non-negative Ricci curvature, showing that complete stable minimal surfaces in such manifolds are either totally geodesic planes or totally geodesic cylinders. The corresponding result is false for general metrics on R. Anderson has obtained examples of complete stable minimal surfaces in hyperbolic 3-space, using the techniques of geometric measure theory [A]. However Anderson's methods do not specify the topological type of these examples. The main theorem of this paper will enable the construction of complete stable minimal surfaces which are not totally geodesic in a wide range of 3-manifolds. In fact the surfaces constructed will be area minimizing on any compact subset, a much stronger condition than stability. In hyperbolic space, we will obtain stable, embedded minimal surfaces whose limit sets are the entire sphere at infinity. The theorem is also applied to construct complete stable, embedded minimal surfaces which are not toally geodesic in another of the geometries on R, in which R is metrically the product of hyperbolic 2-space and the reals. Finally two examples are given of non-totally geodesic area minimizing minimal surfaces in non-simply connected 3-manifolds. We will work in the category of smooth manifolds and maps. An immersed surface is minimal if the first variation of area is zero on any compact subset of F. A minimal surface F is stable if the second variation of area is non-negative on any compact subset of F. A minimal surface F is area minimizing if any compact subsurface Fλ of F minimizes area in

Complete area minimizing minimal surfaces which are not totally geodesic