arXiv subjects

Franciele Conrado

Publications and source records attributed to Franciele Conrado.

6 recordsLinked to original sources

arXiv2026

Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

Let $(M, g, f, τ)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2τ}$ and let $R$ denote its scalar curvature. For a confined function $V$ on $M$, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator $-Δ+\frac{R}{4}+V$, expressed in terms of an integral quantity involving $V$ and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's $μ$-functional. We also study the drifted Schrödinger operator $-Δ_f+V$ on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if $V$ is affine.

math.DG↗

arXiv2025

An eigenvalue estimate for self-shrinkers in a Ricci shirinker

In this paper, we study the drifted Laplacian $Δ_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of $Δ_f$ is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of $Δ_f$ when the hypersurface is an embedded $f$-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.

math.DG↗

arXiv2023

The Wasserstein distance for Ricci shrinkers

Let $(M^n,g,f)$ be a Ricci shrinker such that $\textrm{Ric}_f=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4π)^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_pM$, namely the Gaussian measure $γ$ and the measure $\overlineν$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_0$ between the measures $\overlineν$ and $γ$, and which also elucidates the rigidity implications resulting from this estimate.

math.DG↗

arXiv2023

Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

In this work, we study the rigidity problem for the logarithmic Sobolev inequality on a complete metric measure space $(M^n,g,f)$ with Bakry-Émery Ricci curvature satisfying $Ric_f\geq \frac{a}{2}g$, for some $a>0$. We prove that if equality holds then $M$ is isometric to $Σ\times \mathbb{R}$ for some complete $(n-1)$-dimensional Riemannian manifold $Σ$ and by passing an isometry, $(M^n,g,f)$ must split off the Gaussian shrinking soliton $(\mathbb{R}, dt^2, \frac{a}{2}|.|^2)$. This was proved in 2019 by Ohta and Takatsu. In this paper, we prove this rigidity result using a different method.

math.DG↗

arXiv2021

Disks area-minimizing in mean convex Riemannian $n$-manifolds

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{AMB} to higher dimensions.

math.DG↗

arXiv2019

Topological obstructions to nonnegative scalar curvature and mean convex boundary

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex boundary. For example, we show that the manifold $(T^{n-2}\times Σ)\# N$, where $Σ$ is a compact, connected and orientable surface which is not a disk or a cylinder and $N$ is a closed $n$-dimensional manifold, does not admit a metric of non-negative scalar curvature and mean convex boundary, and the manifold $(I\times T^{n-1})\#N$, where $I=[a,b]$, does not admit a metric of positive scalar curvature and mean convex boundary.

math.DG↗