https://neckpinch.com/authors/matheus-andrade-ribeiro-de-moura-horacio/New arXiv papers by Matheus Andrade Ribeiro de Moura Horácio, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2611.02740/arxiv:2611.02740Mon, 02 Nov 2026 00:00:00 GMTMatheus Andrade Ribeiro de Moura Horácio — We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.Ricci solitonsCurvature pinching4-manifoldsScalar curvatureStability & uniqueness https://neckpinch.com/papers/2610.11935/arxiv:2610.11935Thu, 08 Oct 2026 00:00:00 GMTMatheus Horácio, Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete nonsteady gradient Kähler-Ricci soliton satisfying Ric+∇² f=λg, λ≠ 0. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as N^(2k)×ℂ^(m-k), where N^(2k) is Kähler-Einstein with Ric_(g_N)=λg_N. In the shrinking case the quotient is trivial; in the normalization λ=1/2 one has R≡ k. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if 𝓛_∇fRic is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching 0≤Ric≤λg forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsCurvature pinchingKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://neckpinch.com/papers/2610.11209/arxiv:2610.11209Thu, 08 Oct 2026 00:00:00 GMTMatheus Andrade Ribeiro de Moura Horácio — 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.Ricci solitonsScalar curvatureRotational symmetry & warped products https://neckpinch.com/papers/2609.29359/arxiv:2609.29359Thu, 24 Sep 2026 00:00:00 GMTMatheus Andrade Ribeiro de Moura Horácio — 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.Ricci solitonsKähler–Ricci solitonsKähler geometryScalar curvatureStability & uniqueness https://neckpinch.com/papers/2510.13057/arxiv:2510.13057Wed, 15 Oct 2025 00:00:00 GMTValter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos — 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.Ricci solitonsAlmost & η-Ricci solitons4-manifoldsRotational symmetry & warped productsEigenvalues & spectrum https://neckpinch.com/papers/2510.11939/arxiv:2510.11939Mon, 13 Oct 2025 00:00:00 GMTValter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos — 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 ≥ 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≥5. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.Ricci solitonsEigenvalues & spectrum