A new proof for the existence of rotationally symmetric gradient Ricci solitons

Shu-Yu Hsu

Paper · May 9, 2021

A new proof for the existence of rotationally symmetric gradient Ricci solitons

Shu-Yu Hsu

math.DGarXiv:2105.03805v4

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Abstract

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n + 1$ , $2 \leq n \leq 4$ , with metric $g = \frac{d a ^{2}}{h ( a ^{2} )} + a^{2} d σ$ for some function $h$ where $d σ$ is the standard metric on the unit sphere $S^{n}$ in $R^{n}$ . More precisely for any $λ \geq 0$ , $2 \leq n \leq 4$ and $μ_{1} \in R$ , we prove the existence of unique solution $h \in C^{2} ((0, \infty)) \cap C^{1} ([0, \infty))$ 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 $h (0) = 1$ , $h_{r} (0) = μ_{1}$ . We also prove the existence of unique analytic solution of the about equation on $[0, \infty)$ for any $λ \geq 0$ , $n \geq 2$ and $μ_{1} \in R$ . Moreover we will prove the asymptotic behaviour of the solution $h$ for any $n \geq 2$ , $λ \geq 0$ and $μ_{1} \in R ∖ {0}$ .

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Submitted: May 9, 2021
Last revised: Apr 6, 2024 (v4)
Subjects: math.DG, math.AP

Comments: 33 pages. 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)

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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.

A new proof for the existence of rotationally symmetric gradient Ricci solitons

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