Rigidity of Gradient Shrinking Ricci Solitons with Bach-like Flatness and Related Variational Formulas
James Siene
TL;DR
This work studies rigidity for 4-dimensional complete gradient shrinking Ricci solitons under flatness of Bach-like tensors, generalizing the Bach tensor as $\boldsymbol{\mathfrak{B}}_{ij} = \alpha U_{ij} + \beta V_{ij}$. By developing integral identities with ΔR=0 and using weighted measures $e^{-f}$ (and $e^{-c f}$), the authors relate the vanishing of $\boldsymbol{\mathfrak{B}}_{ij}$ to Cotton-type data via $D_{ijk}$ and to the scalar curvature $R$, showing that outside the Bach line ($\beta \neq \alpha/3$) the vanishing of $\boldsymbol{\mathfrak{B}}_{ij}$ forces the soliton to be Einstein or Gaussian. They treat both $V_{ij}=0$ and $U_{ij}=0$ cases, extend the analysis outside and inside a parameter cone, and provide a variational interpretation of $U$ and $V$ as gradients of a two-parameter family of quadratic curvature functionals, with explicit first and second variation formulas. The results deepen the connection between conformal geometry and soliton rigidity, offering a robust framework to deduce geometric classification from curvature-functionals obstructions.
Abstract
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, except along the Bach line, 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. 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.