Some new developments of realization of surfaces into $R^3$

TL;DR

This survey reviews recent progress in embedding surfaces into three-dimensional space, covering local isometric embeddings, the Weyl problem, negatively curved surfaces, and boundary value problems.

Contribution

It compiles and discusses recent developments and results in the realization of surfaces into R^3 over the past decade.

Findings

Results related to the Schlaffli-Yau conjecture are reviewed.

Developments on Weyl problem for positive curvature are discussed.

Existence results for negatively curved surfaces into R^3 are presented.

Abstract

This paper intends to give a brief survey of the developments on realization of surfaces into $ R^3$ in the last decade. As far as the local isometric embedding is concerned, some results related to the Schlaffli-Yau conjecture are reviewed. As for the realization of surfaces in the large, some developments on Weyl problem for positive curvature and an existence result for realization of complete negatively curved surfaces into $ R^3$, closely related to Hilbert-Efimov theorem, are mentioned. Besides, a few results for two kind of boundary value problems for realization of positive disks into $R^3$ are introduced.