https://neckpinch.com/authors/james-siene/New arXiv papers by James Siene, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2511.05774/arxiv:2511.05774Sat, 08 Nov 2025 00:00:00 GMTJames Siene — 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.Ricci solitons4-manifoldsStability & uniqueness https://neckpinch.com/papers/2307.01882/arxiv:2307.01882Tue, 04 Jul 2023 00:00:00 GMTJames Siene — The Bach tensor is classically defined in dimension 4, and work from J. Bergman and others shows that B = 1/2U + 1/6V where U and V are more basic 2-tensors, which are symmetric, divergence-free, algebraically independent, and quadratic in the Riemann tensor. In this paper, we extend H.-D. Cao and Q. Chen's results for Bach-flat gradient shrinking Ricci solitons to solitons with B = αU + βV = 0.Ricci solitons4-manifolds