Ricci Solitons in Dimensions $4$ and Higher

Ricci Solitons in Dimensions $4$ and Higher offers a detailed account of recent developments of Ricci solitons-self-similar solutions to the Ricci flow equation-which play a central role in modeling the formation of singularities of the flow. Building on the foundational work of Hamilton and Perelman and the recent advances of Bamler, Brendle, and others, this book focuses on the rich and technically demanding theory of these solutions. With special attention to dimension $4$-where potential applications to the topology of smooth 4-manifolds are most promising-the authors present key results, open problems, and new perspectives on the structure and asymptotic behavior of complete noncompact solitons, the case of greatest significance to singularity analysis. The volume offers a systematic and research-oriented reference for ongoing work in geometric analysis, covering both foundational material and specialized topics. Areas of focus include curvature growth and decay, bounds on the number of topological ends, asymptotically conical and asymptotically cylindrical solitons, volume growth, and applications of Bamler's theory. Written for graduate students and researchers in differential geometry, geometric analysis, and mathematical physics, the book is accessible to readers with a solid background in Riemannian geometry and partial differential equations. While self-contained in its core exposition, it serves as both a technical resource and an invitation to contribute to the study of Ricci flow in dimensions $ 4$ and higher.

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