Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

Kin Ming Hui

Paper · Jul 29, 2021

Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

Kin Ming Hui

math.DGarXiv:2107.13685v3

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Abstract

By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g = \frac{d a ^{2}}{h ( a ^{2} )} + a^{2} g_{S^{n}}$ for some function $h$ where $g_{S^{n}}$ is the standard metric on the unit sphere $S^{n}$ in $R^{n}$ for any $n \geq 2$ . More precisely for any $λ \geq 0$ and $c_{0} > 0$ , we prove that there exist infinitely many solutions $h \in C^{2} ((0, \infty); R^{+})$ for the equation $2 r^{2} h (r) h_{r r} (r) = (n - 1) h (r) (h (r) - 1) + r h_{r} (r) (r h_{r} (r) - λ r - (n - 1))$ , $h (r) > 0$ , in $(0, \infty)$ satisfying $r \to 0 lim r^{n - 1} h (r) = c_{0}$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.

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Submitted: Jul 29, 2021
Last revised: Mar 15, 2024 (v3)
Subjects: math.DG, math.AP
Journal: Can. Math. Bull. 67 (2024) 842-859
DOI: 10.4153/S0008439524000237

Comments: typo corrected

Ricci solitons Rotational symmetry & warped products

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

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