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On the existence and classification of $k$-Yamabe gradient solitons

Maria Fernanda Espinal, Mariel Sáez

Abstract

In this paper we classify rotationally symmetric conformally flat admissible solitons to the $k$-Yamabe flow, a fully non-linear version of the Yamabe flow. For $n\geq 2k$ we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For $n<2k$ we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.

On the existence and classification of $k$-Yamabe gradient solitons

Abstract

In this paper we classify rotationally symmetric conformally flat admissible solitons to the -Yamabe flow, a fully non-linear version of the Yamabe flow. For we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.

Paper Structure

This paper contains 31 sections, 34 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. PDE formulation of $k$-Yamabe solitons and proof of Theorem \ref{['PDE_formulation']}
  4. PDE formulation in terms of the eigenvalues of $A_{g_{u}}$ in the radial case
  5. Admissibility of solutions
  6. ODE Analysis
  7. Derivation of the phase-plane system
  8. Critical points of System \ref{['system']}
  9. Choice of orbits
  10. Analysis of critical points of system \ref{['system']}
  11. The fixed point argument for existence at the origin
  12. Global existence of orbits
  13. Existence of orbits in the case $n>2k$
  14. $k$-Yamabe expander soliton $\rho<0$.
  15. $k$-Yamabe steady solitons $\rho=0$.
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $g_{u}$ defined by Then $g_{u}$ is a conformally flat rotationally symmetric $k$-Yamabe gradient soliton with $\sigma_{k}(g_{u})>0$ if and only if $u$ is a smooth radial solution to the elliptic equation where $\theta$ is a parameter that satisfies $2\theta+\rho>0$. Equation elliptic_equation equivalent to where $T_{k-1}$ is the $(k-1)$-th Newton tensor evaluated on the Schouten tensor $g_{

Figures (2)

  • Figure 4.1: Admissible regions in terms of $\rho$ and $\theta$ when $n>2k$.
  • Figure 5.1:

Theorems & Definitions (64)

  • Theorem 1.1: PDE formulation of $k$-Yamabe gradient solitons
  • Theorem 1.2: Existence of radial $k$-Yamabe gradient solitons for $n\geq 2k$
  • Theorem 1.3: Asymptotic behavior
  • Theorem 1.4: Non-existence of radial $k$-Yamabe gradient solitons for $n<2k$
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3: Lemma 4.4 in del2005singular
  • proof : Proof of Theorem \ref{['PDE_formulation']}
  • Claim 1
  • Lemma 3.1
  • ...and 54 more