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Valter Borges

Publications and source records attributed to Valter Borges.

11 recordsLinked to original sources
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.

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arXiv2025

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

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.

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arXiv2025

On gradient $\rho$-Einstein solitons with Bach tensor radially nonnegative

In this paper, we study $n$-dimensional gradient $\rho$-Einstein solitons whose Bach tensor is radially nonnegative. Under this assumption, we show that such $\rho$-Einstein solitons are locally warped products of an interval and an Einstein manifold, provided either $\rho\neq0$ or $\rho=0$ and the soliton is rectifiable. We obtain as a consequence that these solitons must have harmonic Weyl tensor and vanishing Bach tensor. We also finish the classification of complete locally conformally flat steady $\rho$-Einstein solitons and classify these manifolds when their Bach tensor is radially nonnegative and $n\in\{3,4\}$.

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arXiv2023

Topology at infinity of complete gradient Schouten solitons

In this paper, we study ends of complete gradient non-trivial Schouten solitons. Without any additional assumptions, we show the shrinking ones have finitely many ends, and the expanding ones are connected at infinity. We also provide information regarding the non-parabolicity of the ends in the shrinking setting, and on the behavior of the potential function and volume growth in the expanding case.

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arXiv2023

Ends of shrinking gradient $\rho$-Einstein solitons

We prove that all ends of a gradient shrinking $\rho$-Einstein soliton are $\varphi$-non-parabolic, provided $\rho$ is nonnegative and the soliton has bounded and nonnegative scalar curvature, where the weight $\varphi$ is a negative multiple of the potential function. We also show these solitons are connected at infinity for $\rho \in\left[0,1/2(n-1)\right)$, $n\geq4$, and a suitable bound for the scalar curvature.

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arXiv2022

On noncompact warped product Ricci solitons

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.

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arXiv2021

On Einstein hypersurfaces of $I\times_{f}\mathbb{Q}^{n}(c)$

In this paper, we investigate Einstein hypersurfaces of the warped product $I\times_{f}\mathbb{Q}^{n}(c)$, where $\mathbb{Q}^{n}(c)$ is a space form of curvature $c$. We prove that $M$ has at most three distinct principal curvatures and that it is locally a multiply warped product with at most two fibers. We also show that exactly one or two principal curvatures on an open set imply constant sectional curvature on that set. For exactly three distinct principal curvatures this is no longer true, and we classify such hypersurfaces provided it does not have constant sectional curvature and a certain principal curvature vanishes identically.

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arXiv2021

Rigidity of Bach-flat gradient Schouten solitons

In this paper we show that a complete Schouten soliton whose Ricci tensor has at most two eigenvalues at each point is rigid. This allows the classification of both shrinking and expanding Bach-flat Schouten solitons for $n\geq$ 4. When $n=3$ we are able to conclude rigidity under a more general condition, namely when the Bach tensor is divergence free. These results imply rigidity of locally conformally flat Schouten solitons for $n\geq3$.

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arXiv2021

On complete gradient Schouten solitons

In this paper, we prove an optimal inequality between the potential function of a complete Schouten soliton and the norm of its gradient. We also prove that these metrics have bounded scalar curvature of defined sign. As an application, we prove that the potential function of a Shrinking Schouten soliton grows linearly and provide optimal estimates for the growth of the volume of geodesic balls, without any further assumption.

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arXiv2017

Ricci Almost Solitons on semi-Riemannian Warped Products

We characterize Ricci almost solitons on semi-Riemannian warped products, considering the potential function to depend on the fiber or not. We show that the fiber is necessarily an Einstein manifold. As a consequence of our characterization we prove that when the potential function depends on the fiber, if the gradient of the warping function does not act by translations then the base and the warped product are also Einstein manifolds. Moreover, we show the existence of conformal vector fields on the base, the fiber and on the warped product. Assuming completeness of the warped product we provide a classification of such manifolds. When the potential function depends on the fiber and the gradient of the warping functions is an improper vector field, we show that the base is a Brinkmann space and the fiber is Ricci flat. We use the characterization also to prove that the potential function of a complete Ricci soliton depends only on the base.

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