Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some configuration results for area-minimizing cones

Yongsheng Zhang

TL;DR

This paper develops a general framework for generating regular area-minimizing cones in Euclidean spaces by taking cones over minimal products of closed minimal submanifolds in spheres. Central to the approach is Lawlor's curvature criterion, which reduces the problem to normal-radius and second- fundamental-form estimates for the links, and the introduction of vanishing-angle bounds θc and θF to certify area-minimality. The main result shows that the cone over the minimal product of sufficiently many copies of any given closed minimal submanifold is area-minimizing, yielding a vast class of cones with high dimension. Additional configuration results demonstrate how Type-c cones are preserved under product operations and sphere-extensions, illuminating the structure and abundance of regular area-minimizing cones and linking them to isoparametric submanifolds and focal submanifolds. The work provides both concrete constructions and a unifying perspective on the landscape of area-minimizing cones via minimal-product geometry and calibration-type arguments.

Abstract

We discover some configuration results for area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the submanifold turns out to be area-minimizing.

Some configuration results for area-minimizing cones

TL;DR

This paper develops a general framework for generating regular area-minimizing cones in Euclidean spaces by taking cones over minimal products of closed minimal submanifolds in spheres. Central to the approach is Lawlor's curvature criterion, which reduces the problem to normal-radius and second- fundamental-form estimates for the links, and the introduction of vanishing-angle bounds θc and θF to certify area-minimality. The main result shows that the cone over the minimal product of sufficiently many copies of any given closed minimal submanifold is area-minimizing, yielding a vast class of cones with high dimension. Additional configuration results demonstrate how Type-c cones are preserved under product operations and sphere-extensions, illuminating the structure and abundance of regular area-minimizing cones and linking them to isoparametric submanifolds and focal submanifolds. The work provides both concrete constructions and a unifying perspective on the landscape of area-minimizing cones via minimal-product geometry and calibration-type arguments.

Abstract

We discover some configuration results for area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the submanifold turns out to be area-minimizing.
Paper Structure (13 sections, 25 theorems, 73 equations)

This paper contains 13 sections, 25 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Lawlor's curvature criterion
  3. Minimal product
  4. Estimate on $\alpha^2$ for minimal product.
  5. Normal radius of minimal product
  6. Configuration results
  7. For general closed minimal submanifold.
  8. Key control on normal radius of minimal product construction.
  9. Configuration results in the category of Typc-c Lawlor cones.
  10. Appendix
  11. Proof of Lemma \ref{['tF']}.
  12. Proof of Footnote \ref{['ft']}.
  13. Proof of Remark \ref{['Frk']}.

Key Result

Theorem 1.1

Given embedded closed minimal submanifolds $\{L_i\}$ respectively in Euclidean spheres with $i=1,2,\cdots, n$, every cone over the minimal product $L$ of sufficiently many copies among these $\{L_i\}$ is area-minimizing.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Theorem 1.2.1, Law
  • Lemma 2.3: Proposition 1.4.2, Law
  • Lemma 2.4
  • Definition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 51 more