Notes on Chern's Affine Bernstein Conjecture

TL;DR

This paper surveys recent proofs of Chern's affine Bernstein conjecture, highlighting techniques involving Monge-Ampere equations and providing detailed auxiliary material to aid understanding.

Contribution

It provides a detailed comparison of two recent proofs of Chern's conjecture, including auxiliary details omitted in previous work, and discusses related Monge-Ampere equations.

Findings

Two proofs of Chern's conjecture are explained in detail.

The paper clarifies auxiliary steps in the proofs.

It illustrates recent techniques in Monge-Ampere equations.

Abstract

There were two famous conjectures on complete affine maximal surfaces, one due to E. Calabi, the other to S.S. Chern. Both were solved with different methods about one decade ago by studying the associated Euler-Lagrange equation. Here we survey two proofs of Chern's conjecture in our recent monograph [L-X-S-J], in particular we add some details of the proofs of auxiliary material that were omitted in [L-X-S-J]. We describe the related background in our Introduction. Our survey is suitable as a report about recent developments and techniques in the study of certain Monge-Ampere equations.