arXiv subjects

José Nazareno Vieira Gomes

Publications and source records attributed to José Nazareno Vieira Gomes.

6 recordsLinked to original sources

arXiv2026

Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications

We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold $M$. We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all $\mathcal{C}^r$--Riemannian metrics on $M$ or the set of all $\mathcal{C}^r$--diffeomorphisms on a domain into $M$. In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase

math.DG↗

arXiv2026

On a special class of gradient Ricci solitons

We develop a method for constructing complete gradient Ricci solitons realized as fiber bundles endowed with warped metrics, and we establish necessary and sufficient conditions for their existence. As an application, we present new examples of complete gradient steady and shrinking Ricci solitons obtained via quotients by isometric group actions

math.DG↗

arXiv2025

Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE

We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry-Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results.

math.DG↗

arXiv2024

Compact gradient Einstein-type manifolds with boundary

We deal with rigidity results for compact gradient Einstein-type manifolds with nonempty boundaries. As a result, we obtain new characterizations for hemispheres and geodesic balls in simply connected space forms. In dimensions three and five, we obtain topological characterizations for the boundary and upper bounds for its area.

math.DG↗

arXiv2024

Geometric and analytic results for Einstein solitons

We compute a lower bound for the scalar curvature of a gradient Einstein soliton under a certain assumption on its potential function. We establish an asymptotic behavior of the potential function on a noncompact gradient shrinking Einstein soliton. As a result, we obtain the finiteness of its fundamental group and its weighted volume. We also prove some geometric and analytic results for constructing gradient Einstein solitons that are realized as warped metrics, and we give a few explicit examples.

math.DG↗

arXiv2019

A note on gradient Einstein-type manifolds

In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.

math.DG↗