Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.06344

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

William K. Allard
View original source ↗Read source document ↗Export metadata ↓

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 $ε_1$ 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.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

William K. Allard. 2024-07-08. Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones. https://arxiv.org/abs/2407.06344

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum propagation for Berezin-Toeplitz operators

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin-Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducting quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.

math.DG↗
arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗
arXiv2026

Willmore surfaces in 4-dimensional conformal manifolds

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.

math.DG↗