Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e. Uniqueness of Tangent Cones

Camillo De Lellis, Paul Minter, Anna Skorobogatova

Abstract

We consider an area-minimizing integral current $T$ of codimension higher than 1 ins a smooth Riemannian manifold $Σ$. We prove that $T$ has a unique tangent cone, which is a superposition of planes, at $\mathcal{H}^{m-2}$-a.e. point in its support. In combination with works of the first and third authors, we conclude that the singular set of $T$ is countably $(m-2)$-rectifiable.

The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e. Uniqueness of Tangent Cones

Abstract

We consider an area-minimizing integral current of codimension higher than 1 ins a smooth Riemannian manifold . We prove that has a unique tangent cone, which is a superposition of planes, at -a.e. point in its support. In combination with works of the first and third authors, we conclude that the singular set of is countably -rectifiable.
Paper Structure (63 sections, 76 theorems, 751 equations, 3 figures)

This paper contains 63 sections, 76 theorems, 751 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Comparison with the works of Krummel and Wickramasekera
  3. Preliminaries and notation
  4. Notation
  5. Main decay theorem
  6. Structure of the paper
  7. L2-Linfty Height Bound
  8. Main statements
  9. Proof of Corollary \ref{['c:splitting-0']}
  10. Preliminaries
  11. Oriented and non-oriented tilt-excess
  12. The case N=1
  13. Combinatorial lemmas
  14. Well-separated case
  15. Proof of tilt-excess estimate
  16. ...and 48 more sections

Key Result

Theorem 1.1

Let $T$ be an $m$-dimensional area-minimizing integral current in a $C^{3,\kappa_0}$ complete Riemannian manifold of dimension $m+\bar{n}\geq m+2$, with $\kappa_0>0$. Then, $\mathrm{Sing} (T)$ is $(m-2)$-rectifiable and $T$ has a unique tangent cone at $\mathcal{H}^{m-2}$-a.e. $q\in \mathrm{Sing} (T

Figures (3)

  • Figure 1: An illustration of the Whitney decomposition of $R$ (illustrated on $\alpha_1$). The subspace $V$ is represented by the thick line joining the two planes, with a representation cube $L$ showing, with the corresponding sets $R(L)$ (in red) and $L_i$ (in blue). The set $R(L)$ is obtained by rotating the given cube $L_1$ around $V$ in the ambient $(m+n)$-dimensional space, a portion of which is shown.
  • Figure 2: An example of a possible labeling of the cubes and of a corresponding subdivision of $R$. The outer region is white, while the central region is lightly shadowed and the inner region is shadowed. The labels $o$, $c$, and $in$ identify rotationally invariant sets $R (L)$ corresponding to cubes $L$ which are, respectively, outer, central, and inner cubes. Note that descendants of inner cubes are not inner cubes, even though the corresponding rotationally invariant regions are included in the inner region.
  • Figure 3: An illustration of the fattening of some $0$-cells, $1$-cells and $2$-cells, used in the extension algorithm to find a coherent approximation.

Theorems & Definitions (141)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Excess Decay Theorem
  • Theorem 3.2: $L^\infty$ and tilt-excess estimates
  • Corollary 3.3
  • Proposition 4.1
  • ...and 131 more