arXiv subjects

Alcides de Carvalho

Publications and source records attributed to Alcides de Carvalho.

8 recordsLinked to original sources

arXiv2026

Integral Identities and Rigidity of Generalized $m$-Quasi-Einstein Manifolds

We establish differential and integral identities for closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,λ)$. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. Afther this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition and, in particular, proves a recently Colling--Dunajski conjecture at $m=-2$ when $λ\leq0$ is constant. An identity for the drift laplacian gives a new proof of the previously known triviality result for $m\leq-n$ and extends it to generalized $m$-quasi-Einstein manifolds. In the constant-$λ$ case, we then derive a differential identities for twisted operators that singles out the value $m=-4$. Combined with spectral and maximum-principle arguments, this yields triviality for $λ\leq0$, thereby proving the Colling--Dunajski conjecture at $m=-4$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.

math.DG↗

arXiv2026

On Generalized Quasi-Einstein Manifolds

In this paper, we study generalized $m$-quasi-Einstein $(M^n,g,X,λ)$ under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on $m$ and $λ$, including triviality results. We also revisit a recent theorem of Ghosh \cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and proof. Finally, we establish rigidity results for manifolds with geodesic potential vector fields.

math.DG↗

arXiv2026

Index estimates for harmonic Gauss maps

Let $Σ$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within the Lie algebra of $\mathbb{G}$. We prove that the energy index of the Gauss map of $Σ$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.

math.DG↗

arXiv2025

Ruled Ricci surfaces and curves of constant torsion

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the helicoid as the only surface of this kind that admits a parametrization with plane line of striction, and as the only with constant mean curvature.

math.DG↗

arXiv2024

On the stability of free boundary minimal submanifolds in conformal domains

Given a $n$-dimensional Riemannian manifold with non-negative sectional curvatures and convex boundary, that is conformal to an Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension $2\leq k\leq n-2$, provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvatures are strictly positive.

math.DG↗

arXiv2022

Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Balls

In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved by A. Fraser and M. Li \cite{FRLI}. As a consequence, we obtain the two-piece property for free boundary minimal hypersurfaces in the unit ball: every equatorial disk divides any compact minimal hypersurface with free boundary in the unit ball in two connected pieces.

math.DG↗

arXiv2021

Minimal Kähler submanifolds in product of space forms

In this article, we study minimal isometric immersions of Kähler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a Kähler manifold into those spaces and we prove that the only ones into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{H}^{m-1}\times \mathbb{R}$ are the minimal isometric immersions of Riemannian surfaces. Futhermore, we show that the existence of a minimal isometric immersion of a Kähler manifold $M^{2n}$ into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{S}^{m-k}\times \mathbb{H}^k$ imposes strong restrictions on the Ricci and scalar curvatures of $M^{2n}$. In this direction, we characterise some cases as either isometric immersions with parallel second fundamental form or anti-pluriharmonic isometric immersions.

math.DG↗

arXiv2019

Real Kähler Submanifolds in Codimension $6$

We show that a real Kähler submanifold in codimension $6$ is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity.

math.DG↗