Structure at infinity for shrinking Ricci solitons

Munteanu, Ovidiu; Wang, Jiaping

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 891-925.

Cet article concerne principalement la structure à l'infini pour gradient solitons rétrécis de Ricci. Il est montré que pour un tel soliton avec courbure bornée, si le cylindre rond $ℝ\times {𝕊}^{n-1}/\Gamma$ se produit comme une limite pour une séquence de points convergeant à l'infini le long d'une extrémité, alors l'extrémité est asymptotique au même cylindre rond à l'infini. Le résultat est appliqué pour obtenir des résultats structurels à l'infini pour gradient solitons de Ricci de dimension quatre. On sait déjà que ces solitons avec courbure scalaire proche de zéro à l'infini doivent être asymptotiques à un cône. Dans le cas où la courbure scalaire est délimitée par le bas par une constante positive, nous concluons que le long de chaque extrémité le soliton est asymptotique à un quotient de $ℝ\times {𝕊}^3$ ou converge vers un quotient de ${ℝ}^2\times {𝕊}^2$ le long de chaque courbe intégrale du champ de vecteur de gradient de la fonction potentielle.

This paper concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $ℝ\times {𝕊}^{n-1}/\Gamma$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic at infinity to the same round cylinder. This result is applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It was previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of $ℝ\times {𝕊}^3$ or converges to a quotient of ${ℝ}^2\times {𝕊}^2$ along each integral curve of the gradient vector field of the potential function.

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