Abstract
We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylinders $\mathbb{S}^k \times \mathbb{R}^{n-k}$ for $k \geq 2$ along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when $k =n-1$) to its $\mathbb{Z}_2$-quotient.
Funding Statement
The first author was supported in part by Simons Foundation grant #359335.
The second author was supported in part by NSF grants DMS-2018221 (formerly DMS-1406240) and DMS-2018220, an Alfred P. Sloan research fellowship, and the office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
Citation
Brett Kotschwar. Lu Wang. "A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons." J. Differential Geom. 126 (1) 215 - 295, 1 January 2024. https://doi.org/10.4310/jdg/1707767338
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