Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detect duality obstruction of calibrations in smooth category

Yongsheng Zhang

Abstract

This paper consists of three parts: (a) exhibit a new gluing result which can dramatically simplify extensions of calibration pairs; (b) observe that every Lawlor cone can support coflat calibrations singular only at the origin; (c) show that there exist many Lawlor cones which cannot support any smooth calibrations. As an application, we extend our previous work on detecting duality obstruction of calibrations in the smooth category.

Detect duality obstruction of calibrations in smooth category

Abstract

This paper consists of three parts: (a) exhibit a new gluing result which can dramatically simplify extensions of calibration pairs; (b) observe that every Lawlor cone can support coflat calibrations singular only at the origin; (c) show that there exist many Lawlor cones which cannot support any smooth calibrations. As an application, we extend our previous work on detecting duality obstruction of calibrations in the smooth category.

Paper Structure

This paper contains 18 sections, 21 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basics of calibrations
  3. Definitions and basic properties.
  4. Four lemmas.
  5. New gluing result for the extension of calibration pairs
  6. A new gluing result.
  7. Simplified proof of Theorem \ref{['conecal']}
  8. Every Lawlor cone supports calibrations singular only at the origin
  9. Introduction to Lawlor's curvature criterion.
  10. Proof of Theorem \ref{['T1']}.
  11. A shortcut for the realization result in z.
  12. Minimal product and a proof of Theorem \ref{['T2']}
  13. Minimal product.
  14. Proof of Theorem \ref{['T2']}
  15. Detect duality obstruction in higher codimension
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $L$ be a closed embedded submanifold in $\mathbb S^N\subset \mathbb R^{N+1}$. If the cone $C(L)=\{tp\, : \, t\in(0,\infty),\, p\in L\}$ can be shown area-minimizing by Lawlor's criterion, then $C(L)$ supports coflat calibrations singular only at the origin.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 37 more