arXiv subjects

J. N. V. Gomes

Publications and source records attributed to J. N. V. Gomes.

3 recordsLinked to original sources

arXiv2019

Gradient Ricci almost soliton warped product

We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our construction. We also prove a rigidity result for a gradient Ricci soliton Riemannian product in the class of gradient Ricci almost soliton warped products under some natural geometric assumptions on the warping function.

math.DG↗

arXiv2019

Hadamard type variation formulas for the eigenvalues of the $η$-Laplacian and applications

We consider an analytic family of Riemannian metrics on a compact smooth manifold $M$. We assume the Dirichlet boundary condition for the $η$-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all $C^r$ Riemannian metrics $\mathcal{M}^r$ on $M$, all eigenvalues of the $η$-Laplacian are generically simple, for $2\leq r< \infty$. This implies the existence of a residual set of metrics in $\mathcal{M}^r$, which makes the spectrum of the $η$-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions $η$ in the space $\mathcal{F}^r$ of all $C^r$ functions on $M$, which makes again the spectrum of the $η$-Laplacian simple, for $2\leq r< \infty$. Besides, we give a precise information about the complementary of these residual sets, as well as about the structure of the set of deformations of a Riemannian metric (respectively of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard type formulas for the eigenvalues of the $η$-Laplacian in this case. We also establish generic properties of eigenvalues in this context.

math.DG↗

arXiv2017

On the construction of gradient Ricci soliton warped product

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and sufficient condition for constructing a gradient Ricci soliton warped product. As an application, we present a class of expanding Ricci soliton warped product having as a fiber an Einstein manifold with non-positive scalar curvature. We also discuss some obstructions to this construction, especially in the case when the base of the warped product is compact.

math.DG↗