2025/11/08 by Siene, James
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · doi:10.48550/arxiv.2511.05774
openalex publication_date 2025/11/08 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
The classical Bach tensor in four dimensions can be expressed as a linear combination of two independent, symmetric, divergence-free, quadratic-in-curvature tensors U and V. Several classification results for gradient-shrinking Ricci solitons have been obtained under the assumption that the Bach tensor vanishes. We define a Bach-like tensor to be any other linear combination of U and V. We prove that within a certain cone of parameters, the vanishing of a Bach-like tensor forces a four-dimensional complete gradient-shrinking Ricci soliton to be either Einstein or isometric to the Gaussian soliton, extending the results of Cao--Chen (2013). The special case where U=0 forces fmin∈\0,1,2\, with rigidity holding when fmin=0,2. The remaining case fmin=1 is the central open problem, with a cylinder as the conjectured exceptional geometry. Finally, we show that Bach-like tensors arise as Euler--Lagrange equations of a two-parameter family of quadratic curvature functionals and compute the corresponding first and second variation formulas.