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

TL;DR

This paper extends classification results for four-dimensional gradient-shrinking Ricci solitons by analyzing the vanishing of a generalized Bach-like tensor, and explores related variational formulas for quadratic curvature functionals.

Contribution

It introduces a broader class of Bach-like tensors, proves rigidity results under their vanishing, and derives variational formulas for associated curvature functionals.

Findings

Vanishing of a Bach-like tensor implies the soliton is Einstein or Gaussian.

Special cases determine the minimum of the potential function, with rigidity in certain cases.

Derived first and second variation formulas for quadratic curvature functionals.

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 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.