Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dimension of the singular set for $2$-valued stationary Lipschitz graphs

Jonas Hirsch, Luca Spolaor

TL;DR

This work determines the optimal size of the singular set for 2-valued Lipschitz graphs stationary for the area, proving a codimension-1 bound on Sing$(f)$ in the general case and a sharper bound in the presence of regular-branching with multiplicity two. The authors adapt Almgren’s strategy to the stationary setting by developing ${ m A}_Q$-generalized gradient Young measures, proving higher integrability for the Dirichlet energy, and constructing a strong Lipschitz approximation with superlinear excess via Almgren-type approximation. A center-manifold/normal-approximation framework and a refined blow-up/capacity analysis are then employed to establish a contradiction argument that yields the dimension bound, without assuming area-minimization. The results advance understanding of the singular structure of stationary varifolds in the multiplicity-two, codimension-one regime and provide tools that may extend to stationary stable varifolds and related multivalued-graph problems.

Abstract

We prove that the singular set of a $2$-valued Lipschitz graph that is stationary for the area is of codimension $1$.

Dimension of the singular set for $2$-valued stationary Lipschitz graphs

TL;DR

This work determines the optimal size of the singular set for 2-valued Lipschitz graphs stationary for the area, proving a codimension-1 bound on Sing in the general case and a sharper bound in the presence of regular-branching with multiplicity two. The authors adapt Almgren’s strategy to the stationary setting by developing -generalized gradient Young measures, proving higher integrability for the Dirichlet energy, and constructing a strong Lipschitz approximation with superlinear excess via Almgren-type approximation. A center-manifold/normal-approximation framework and a refined blow-up/capacity analysis are then employed to establish a contradiction argument that yields the dimension bound, without assuming area-minimization. The results advance understanding of the singular structure of stationary varifolds in the multiplicity-two, codimension-one regime and provide tools that may extend to stationary stable varifolds and related multivalued-graph problems.

Abstract

We prove that the singular set of a -valued Lipschitz graph that is stationary for the area is of codimension .
Paper Structure (29 sections, 28 theorems, 241 equations)

This paper contains 29 sections, 28 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Strategy and main new contributions
  3. About our assumptions
  4. Organization of the paper
  5. Some notations
  6. Acknowledgments
  7. ${\mathcal{A}}_Q$ generalized gradient Young measures
  8. Proof of Proposition \ref{['le:weakclosed']}
  9. Proof of Proposition \ref{['p:elementarylimit']}
  10. Linear theory for measure solutions
  11. Basic properties of inner and outer measure solutions
  12. Regularity: proof of Theorem \ref{['thm:UCformeas']}
  13. Proof of Propositions \ref{['prop:higofmeas']} and \ref{['prop:regformeas']}
  14. Compactness: proof of Theorem \ref{['thm:compactness']}
  15. Almgren's Strong approximation
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $f\colon \Omega \to {\mathcal{A}}_{2}(\mathbb{R}^n)$, with $\Omega\subset \mathbb{R}^m$ open, be a Lipschitz map such that $\mathbf{G}_f$ is a stationary varifold. Then $\dim (\mathrm{Sing}(f)\cap \Omega)\leq m-1$ and all the points in $\mathrm{Reg}(f)$ have either multiplicity $1$ or $2$. Furth

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2: Dimension of the singular set
  • Corollary 1.3: Multiplicity $2$ branching set for codimension $1$ stationary stable varifolds with no triple junctions
  • Remark 1.4: Graphicality and multiplicity $2$
  • Remark 1.5: On the stationariety of multivalued graphs
  • Definition 2.1: ${\mathcal{A}}_Q$ generalized Young measures
  • Proposition 2.2: $\mathcal{Y}^Q$ is weakly closed
  • Definition 2.3: Elementary measures
  • Definition 2.4: ${\mathcal{A}}_Q$ generalized gradient Young measures
  • Proposition 2.5: Elementary properties of gradient measures
  • ...and 46 more