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Open $3$-manifolds with non negative Ricci curvature in a spectral or integral sense

Gilles Carron

Abstract

We show that if a complete Riemannian $3-$manifold has $L^{\frac 32}-$ integrable Ricci curvature, satisfies a Sobolev inequality and has a non negative Ricci curvature in a spectral sense, then it is diffeomorphic to $\R^3$.

Open $3$-manifolds with non negative Ricci curvature in a spectral or integral sense

Abstract

We show that if a complete Riemannian manifold has integrable Ricci curvature, satisfies a Sobolev inequality and has a non negative Ricci curvature in a spectral sense, then it is diffeomorphic to .
Paper Structure (21 sections, 23 theorems, 143 equations)

This paper contains 21 sections, 23 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Geometry after time change
  3. About the metric $\mathop{\mathrm{ \overline{\emph{g}}}}\nolimits$
  4. The scalar curvature of the metric $\mathop{\mathrm{ \overline{\emph{g}}}}\nolimits$
  5. Properties of the gauge
  6. With a Sobolev inequality
  7. Integral bound on the gradient of the Green kernel
  8. $\mu$-volume of $\mathop{\mathrm{ \overline{\emph{g}}}}\nolimits$ geodesic ball
  9. Sobolev inequality for $\mathop{\mathrm{ \overline{\emph{g}}}}\nolimits$
  10. A Vanishing result for the $\upalpha$-genus
  11. Proof of the corollaries
  12. Proof of Theorem \ref{['Vanishing']}
  13. Construction of the Dirichlet to Neuman map
  14. A vanishing result for the cohomology defined by the Dirichlet to Neuman map
  15. Conclusion via Callias index formula
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(M^3,g)$ be a complete Riemannian manifold satisfying the Sobolev inequality Sob and such that $\hbox{Ric}_{\hbox{\tiny{--}}}\in L^{3/2}$. If for some $\lambda>2$, the Schrödinger operator $\Delta-\lambda\hbox{Ric}_{\hbox{\tiny{--}}}$ is non negative, then $M^3$ is diffeomorphic to $\mathbb R^3

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Corollary 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 20 more