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On the stability of the Yamabe invariant of $S^3$

Liam Mazurowski, Xuan Yao

Abstract

Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $Δ_g w + λw^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.

On the stability of the Yamabe invariant of $S^3$

Abstract

Let be a complete, asymptotically flat metric on with vanishing scalar curvature. Moreover, assume that supports a nearly Euclidean Sobolev inequality. We prove that must be close to Euclidean space with respect to the -distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of . More precisely, we show that if such a manifold carries a suitably normalized, positive solution to then must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.
Paper Structure (10 sections, 18 theorems, 149 equations)

This paper contains 10 sections, 18 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Scalar Curvature Stability Results
  3. Main Results
  4. Sketch of Proof
  5. Organization
  6. Acknowledgements
  7. A Willmore type inequality
  8. Isoperimetric Control
  9. Convergence to Euclidean in the $d_p$ metric
  10. Convergence of Conformal Factors

Key Result

Theorem 1

Assume that $(M_i,g_i)$ is a sequence of complete, asymptotically flat 3-manifolds with vanishing scalar curvature. Assume that each $M_i$ is topologically $\mathbb{R}^3$. Further suppose that where $\delta_i\to 0$. Then $M_i$ converges to Euclidean space in the $d_p$ sense for all $p\in(3,\infty)$.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • proof
  • ...and 24 more