The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor

TL;DR

The paper supplies a shorter, frame-free proof that gradient Ricci solitons with harmonic Weyl curvature in dimensions $n\ge4$ have Ricci tensors with at most three distinct eigenvalues. It first establishes a local multiply warped product structure with a 1D base and at most two Einstein fibers, using an arc-length coordinate $s$ along $\nabla f$ and integrable Ricci-eigenspace distributions. A key step is deriving a degree-2 polynomial whose roots are the warp ratios $\xi_i=h_i'/h_i$, which enforces the bound to two fibers and yields the three-eigenvalue limit for the Ricci tensor. This approach circumvents moving frames and strengthens the local-to-global classification framework for such solitons, aligning with Kim's prior results but in a more streamlined fashion.

Abstract

In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension $n \geq 4$ and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension $4$ and subsequently extended to dimensions $n\geq5$. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.

Theorems & Definitions (8)

• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['decompwarpint']}
• proof : Proof of Theorem \ref{['maxxnumeig-INTRO']}