• Corpus ID: 119140719

Lectures on Minimal Surface Theory

@article{White2013LecturesOM,
  title={Lectures on Minimal Surface Theory},
  author={Brian White},
  journal={arXiv: Differential Geometry},
  year={2013}
}
  • B. White
  • Published 15 August 2013
  • Mathematics
  • arXiv: Differential Geometry
An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute. 

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References

SHOWING 1-10 OF 60 REFERENCES

Stable complete minimal surfaces in $R^3$ are planes

. A proof of the statement in the title is given.

The classical theory of minimal surfaces

We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the

Embedded Minimal Surfaces with Finite Topology

We present a synthesis of the situation as it now stands about the various moduli spaces of properly embedded minimal surfaces of finite topology in flat 3-manifolds. This family includes the case of

The maximum principle for minimal varieties of arbitrary codimension

We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an

Genus-one helicoids from a variational point of view

In this paper, we use variational methods to prove existence of a complete, properly embedded, genus-one minimal surface that is asymptotic to a helicoid at infinity. We also prove some new

A local regularity theorem for mean curvature flow

This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1.

An embedded genus-one helicoid.

TLDR
There exists a properly embedded minimal surface of genus one with a single end asymptotic to the end of the helicoid constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces that have genus equal to one in the quotient.

The Boundary Regularity of Minimal Surfaces

In this chapter we deal with the boundary behaviour of minimal surfaces with particular emphasis on the behaviour of stationary surfaces at their free boundaries. This and the following chapter will

Gunther's proof of Nash's isometric embedding theorem

An complete exposition of Matthias Gunther's elementary proof of Nash's isometric embedding theorem.

A Course in Minimal Surfaces

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential
...