https://neckpinch.com/authors/valter-borges/New arXiv papers by Valter Borges, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2510.13057arxiv: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://arxiv.org/abs/2510.11939arxiv: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 https://arxiv.org/abs/2203.06441arxiv:2203.06441Sat, 12 Mar 2022 00:00:00 GMTValter Borges — The goal of this article is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain pde estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.Ricci solitonsEinstein metricsStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsEigenvalues & spectrum