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Lucas Ambrozio

Publications and source records attributed to Lucas Ambrozio.

17 recordsLinked to original sources
arXiv2026

Ambient geometry via min-max widths of embedded circles

We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its 1-width in Urysohn's quantitative dimension theory. In the specific case of the Euclidean plane, we bound from below the classical width of a simple closed curve by its min-max width. Finally, we prove related rigidity results characterizing the round spheres among Zoll spheres.

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arXiv2026

Spheres with minimal equators

We survey the classification of the Riemannian metrics on spheres with respect to which all equators are minimal hypersurfaces, and discuss problems related to these geometries.

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arXiv2025

The width of embedded circles

We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also investigate properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width.

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arXiv2025

Equivariant constructions of spheres with Zoll families of minimal spheres

We construct one-parameter deformations of the Euclidean sphere $\mathbb{S}^n$ inside $\mathbb{R}^{n+1}$ that admit a Zoll family of codimension one embedded minimal spheres, in all dimensions $n\geq 3$. The method of construction is equivariant with respect to the natural actions of the orthogonal group. In particular, we show that the original Zoll spheres of revolution in $\mathbb{R}^3$ have counterparts in the context of minimal surface theory, in all dimensions. We also describe the first examples of metrics on the real projective spaces $\mathbb{RP}^n$, in all dimensions $n \geq 3$, that admit a Zoll family of embedded minimal projective hyperplanes, and which are not isometric to metrics with minimal linear projective hyperplanes. The new constructions are underpinned by equivariant versions of Nash-Moser-Hamilton implicit function theorem, and yield new information even in dimension $n=2$. As an application, we also show that every finite group of the orthogonal group $O(3)$ that does not contain $-Id$ is the isometry group of some (classical) Zoll metric on $\mathbb{S}^2$.

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arXiv2024

Rigidity theorems for the area widths of Riemannian manifolds

The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space.

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arXiv2024

The intrinsic metric of constant mean curvature surfaces and minimal hypersurfaces with free boundary

Ricci-Curbastro established necessary and sufficient conditions for a Riemannian metric on a surface to be the first fundamental form of a minimal immersion of that surface into the Euclidean space. We revisit certain developments arising from his theorem, and propose new versions of these results in the context of the theory of constant mean curvature surfaces in three-dimensional space forms that meet umbilical surfaces orthogonally along their boundary components. Higher dimensional generalisations, inspired by a theorem of do Carmo and Dajczer, are discussed as well.

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arXiv2021

Riemannian metrics on the sphere with Zoll families of minimal hypersurfaces

In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function Theorem in the tame maps setting of Hamilton. This answers a question of Yau on perturbations of minimal hypersurfaces in positive Ricci curvature. We also consider the case of the projective space and characterize those metrics on the sphere with minimal equators.

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arXiv2019

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in $\mathbb{R}^3$ unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.

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arXiv2018

On the two-systole of real projective spaces

We establish an integral-geometric formula for minimal two-spheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional real projective space as the unique metric with the largest possible two-systole among metrics with the same volume in its conformal class.

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arXiv2018

On the min-max width of unit volume three-spheres

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Smith min-max theory.

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arXiv2018

Geometric convergence results for closed minimal surfaces via bubbling analysis

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $\gamma$ is sequentially compact for any $\gamma\geq 1$. Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $m\geq 1$, away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

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arXiv2018

Compactness analysis for free boundary minimal hypersurfaces

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.

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arXiv2017

Index estimates for free boundary minimal hypersurfaces

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser-Schoen and Folha-Pacard-Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed.

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arXiv2016

Comparing the Morse index and the first Betti number of minimal hypersurfaces

By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques-Neves for a wide class of ambient spaces.

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arXiv2015

Compactness of the space of minimal hypersurfaces with bounded volume and p-th Jacobi eigenvalue

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.

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