Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Codimension 2 drawstrings with scalar curvature lower bounds

Demetre Kazaras, Kai Xu

TL;DR

This work constructs drawstring metrics along codimension-2 submanifolds to create lower scalar curvature bounds while collapsing geometry, producing a flexible framework g' that matches the original metric away from a small neighborhood and prescribes a conformal factor on Σ. A central moving-frame analysis yields a scalar curvature estimate showing R_{g'} is bounded below by a positive main term plus controllable errors, and h,u are explicitly built to realize the desired boundary and volume properties. The authors deploy these tools to generate collapse phenomena, uniform distance-function limits, and various stability/instability results for scalar-curvature-related theorems, including the Positive Mass Theorem and Llarull-type area-extremality, while also constructing asymptotically flat examples and exploring conformal-limit realizations. Overall, the paper provides a robust method to approximate or collapse along codimension-2 sets under scalar-curvature constraints, yielding new insights into scalar-curvature stability, limit spaces (pulled string spaces), and the sharpness of existing rigidity results with explicit geometric control.

Abstract

We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of scalar curvature rigidity phenomena, such as Llarull's Theorem and the Positive Mass Theorem.

Codimension 2 drawstrings with scalar curvature lower bounds

TL;DR

This work constructs drawstring metrics along codimension-2 submanifolds to create lower scalar curvature bounds while collapsing geometry, producing a flexible framework g' that matches the original metric away from a small neighborhood and prescribes a conformal factor on Σ. A central moving-frame analysis yields a scalar curvature estimate showing R_{g'} is bounded below by a positive main term plus controllable errors, and h,u are explicitly built to realize the desired boundary and volume properties. The authors deploy these tools to generate collapse phenomena, uniform distance-function limits, and various stability/instability results for scalar-curvature-related theorems, including the Positive Mass Theorem and Llarull-type area-extremality, while also constructing asymptotically flat examples and exploring conformal-limit realizations. Overall, the paper provides a robust method to approximate or collapse along codimension-2 sets under scalar-curvature constraints, yielding new insights into scalar-curvature stability, limit spaces (pulled string spaces), and the sharpness of existing rigidity results with explicit geometric control.

Abstract

We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of scalar curvature rigidity phenomena, such as Llarull's Theorem and the Positive Mass Theorem.
Paper Structure (25 sections, 33 theorems, 245 equations, 2 figures)

This paper contains 25 sections, 33 theorems, 245 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and main theorem
  3. The structure of drawstring metrics
  4. Applications
  5. Collapsing under scalar curvature lower bounds.
  6. Pointwise limit of distance functions.
  7. Stability of the Positive Mass Theorem.
  8. Area instability of Llarull's Theorem.
  9. The main construction and statements
  10. Computation of scalar curvature
  11. Strategy and setup to prove Theorem \ref{['thm-main:scal_estimate']}
  12. Controlling the Lie brackets
  13. Second fundamental form of Σr
  14. The scalar curvature of Σr
  15. The proof of Theorem \ref{['thm-main:scal_estimate']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $n\geqslant3$ and $\Sigma^{n-2}$ be a closed oriented embedded codimension-2 submanifold of an oriented Riemannian $n$-manifold $(M^n,g)$. Given any $\varepsilon>0$ and function $v_0\in C^\infty(\Sigma)$ with $v_0\leqslant0$, there is a radius $r_1<\varepsilon$ and another metric $g'$ on $M$ sat

Figures (2)

  • Figure 1: The cutoff functions that appear when defining $h$ and $w$.
  • Figure 2: The graphs of $h$ and $w$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: collapsing a submanifold
  • Theorem 1.5: partial collapsing
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 60 more