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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.06344 |
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| _version_ | 1866911948760678400 |
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| author | Allard, William K. |
| author_facet | Allard, William K. |
| contents | 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 $ε_1$ through $ε_6$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones Allard, William K. Differential Geometry 49Q20: Variational problems in a geometric measure-theoretic setting 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 $ε_1$ through $ε_6$ 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. |
| title | Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones |
| topic | Differential Geometry 49Q20: Variational problems in a geometric measure-theoretic setting |
| url | https://arxiv.org/abs/2407.06344 |