Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas

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 $f_{\min}\in\{0,1,2\}$, with rigidity holding when $f_{\min}=0,2$. The remaining case $f_{\min}=1$ 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.