Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones

TL;DR

This paper corrects and rewrites significant portions of Allard and Almgren's 1981 work on tangent cones, addressing errors, omissions, and incomplete proofs to ensure mathematical accuracy and clarity.

Contribution

It provides a complete revision of key sections of Allard and Almgren's original paper, fixing errors and filling gaps in proofs and definitions.

Findings

Corrected inequalities 5.3(4),(5) from AA81

Rewritten sections 5.1-5.4 for clarity and completeness

Clarified the origin of the variable z and related definitions

Abstract

The paper referred to in the title is Allard/Almgren 1981 [AA81]. Several months ago Francesco Maggi emailed me saying that the inequalities 5.3(4),(5) of \cite{AA81} were wrong. In fact, as he pointed out, their incorrectness is immediately apparent if one takes $Z=0$ there. Maggi and his coauthor wanted to use these inequalities in their paper \cite{MN}. They were able to obtain a version of these inequalities which suffice for the carrying out the work in \cite{MN}. I started writing this paper in order to provide a version of these inequalities as needed in [AA81]. In thinking about this material I began to realize there were other problems with the paper. As a result I ended up {\em completely rewriting 5.1-5.4 on pages 243-248 of \cite{AA81}}; this rewrite is the contents of this paper. In addition to many annoying misprints many of the needed definitions and proofs in 5.1-5.4 are incomplete or absent. For example, it is not said where $z$ in 5.1(2) comes from; this omission completely surprised me since I remember doing a lot of work to come up with $z$ when \cite{AA81} was being written. Also, much of the necessary material about the reach of a submanifold as in \cite{FE2} is not provided in \cite{AA81}. The table of contents can serve as an index. In particular one sees there where the constants $\epsilon_1$ through $\epsilon_6$ are introduced. This material is extremely technical. One way to navigate this paper would be to start looking at Proposition 9.5 and work backwards.