arXiv subjects

Matheus Andrade Ribeiro de Moura Horácio

Publications and source records attributed to Matheus Andrade Ribeiro de Moura Horácio.

4 recordsLinked to original sources

arXiv2026

On Ricci solitons whose level hypersurfaces have parallel second fundamental form

We study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.

math.DG↗

arXiv2026

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

In this article, we investigate gradient Ricci solitons of dimension $n\geq4$ with harmonic Weyl tensor. We prove the equivalence of several conditions involving the Ricci eigendistributions, the geometry and the intrinsic Ricci tensor of the level sets of the potential function, the local decomposability of the corresponding net, and local multiply warped product representations. Under the local decomposability assumption, we provide a short proof of Kim's bound on the number of distinct Ricci eigenvalues which does not require the use of specialized moving frames. Combining Kim's theorem with our equivalence result, we also obtain the remaining geometric conditions without any additional hypothesis.

math.DG↗

arXiv2026

Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

We prove that every complete nonsteady gradient Kähler-Ricci soliton with constant scalar curvature is rigid. This establishes Cao's rigidity conjecture for Kähler-Ricci solitons in arbitrary dimension and gives the corresponding result for expanding solitons, without additional curvature assumptions. The proof uses a rigidity criterion for gradient Ricci solitons with constant scalar curvature, expressed by the vanishing of the Lie derivative of the Ricci tensor along the soliton vector field. In the Kähler case, this vanishing follows from constant scalar curvature, the closedness of the Ricci form, and the classical real holomorphicity of the soliton vector field.

math.DG↗

arXiv2025

Local structure of gradient almost Ricci solitons with harmonic Weyl tensor

In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.

math.DG↗