arXiv subjects

Letizia Branca

Publications and source records attributed to Letizia Branca.

6 recordsLinked to original sources

arXiv2026

On the Bach tensor and quadratic curvature functionals

We study the critical points of a quadratic functional depending on the gradient of the Bach tensor on Riemannian four-manifolds, which generalize the Bach-flat condition. We show that, on every closed four-manifold, there exists a weak Bach-parallel metric, i.e. a critical point for this functional with respect to conformal variations: in particular, we prove that there exist infinitely many conformal classes which contain a unique minimizer for the functional, up to constant positive rescaling. Next, we analyze the global minima of the functional, i.e. metrics with parallel Bach tensor, relating these metrics to well-known variational problems. Using a version of de Rham's splitting theorem on complete four-manifolds, we provide a classification result for products of surfaces, exploiting the theory of conformal gradient solitons; we also construct a new explicit example of a Bach-flat metric which is neither locally conformally flat nor conformally Einstein and we characterize HCMU metrics on complete surfaces. Finally, we prove an equivalence between the Bach-parallel condition on 4D cylinders and the existence of critical metrics for a well-known quadratic curvature functional in dimension three: in this direction, we also prove a characterization of flat three-manifolds, under some curvature and finite energy assumptions.

math.DG↗

arXiv2026

Some isolation and stability results for Einstein manifolds

We prove new isolation and stability results for Einstein manifolds in a variety of settings. Imposing conditions on the Weyl tensor, we establish new stability criteria for compact, asymptotically hyperbolic (AH) and asymptotically locally Euclidean (ALE) manifolds and an isolation result in the latter setting. For compact Kähler and Sasaki $η$-Einstein manifolds, we provide similar results which involve the Bochner tensor and the contact Bochner tensor, respectively.

math.DG↗

arXiv2025

On the Geometry of $φ$-Static Perfect Fluid Space-Times

In this paper we study the geometry of $φ$-static perfect fluid space-times ($φ$-SPFST, for short). In the context of Einstein's General Relativity, they arise from a space-time whose matter content is described by a perfect fluid in addition to a nonlinear field expressed by a smooth map $φ$ with values in a Riemannian manifold. Considering the Lorentzian manifold $\hat{M}$ in the form of a static warped product, we derive the fundamental equations via reduction of Einstein's Field Equations to the factors of the product. To set the stage for our main results, we discuss the validity of the classical Energy Conditions in the present setting and we introduce the formalism of $φ$-curvatures, which is a fundamental tool to merge the geometry of the manifold with that of the smooth map $φ$. We then present several mathematical settings in which similar structures arise. After computing two integrability conditions, we apply them to prove a number of rigidity results, both for manifolds with or without boundary. In each of the aforementioned results, the main assumption is given by the vanishing of some $φ$-curvature tensors and the conclusion is a local splitting of the metric into a warped product. Inspired by the classical Cosmic No Hair Conjecture of Boucher, Gibbons and Horowitz, we find sharp sufficient conditions on a compact $φ$-SPFST with boundary to be isometric to the standard hemisphere. We then describe the geometry of relatively compact domains in $M$ subject to an upper bound on the mean curvature of their boundaries. Finally, we study non-existence results for $φ$-SPFSTs, both via the existence of zeroes of the solutions of an appropriate ODE and with the aid of a suitable integral formula generalising in a precise sense the well-known Kazdan-Warner obstruction.

math.DG↗

arXiv2025

Some geometric properties of generalized $φ$-vacuum static spaces

In this paper we study the geometry of generalized $φ$-vacuum static spaces, proving estimates for the $φ$-scalar curvature and for the first eigenvalue of the Jacobi operator, and also rigidity under various geometric assumptions; in particular, we prove a result related to the famous Cosmic no-hair conjecture of Boucher, Gibbons and Horowitz.

math.DG↗

arXiv2025

Bach-pinched metrics on closed manifolds

Exploiting the deformation method introduced by Aubin in his seminal work to construct constant negative scalar curvature metrics, we show the existence, on every closed manifold of dimension four, of a metric whose Bach tensor is pinched by the scalar curvature.

math.DG↗

arXiv2024

Rigidity of Einstein manifolds with positive Yamabe invariant

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant \emph{via} the $L^{\frac{n}{2}}$-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions $5$ and $6$.

math.DG↗