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Roney Santos

Publications and source records attributed to Roney Santos.

8 recordsLinked to original sources
arXiv2026

Zoll manifolds with boundary

We introduce and study Zoll manifolds with boundary: compact Riemannian manifolds with smooth boundary such that every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. We first show that all such free boundary geodesics are embedded and have a common length, and that the boundary has at most two connected components. If there are two components, we prove that the manifold is a product of an interval with a closed manifold. When the boundary is connected, we show that the manifold is a tubular neighborhood of a closed embedded submanifold, the "soul", and that the complement of the soul is diffeomorphic to a half-open cylinder over the boundary. We further prove that all free boundary geodesics are maximally degenerate critical points of the energy functional and have the same Morse index, which equals the multiplicity of the unique focal point occurring at the midpoint of each geodesic. The projection from the boundary to the soul is then either a nontrivial two-fold covering or a smooth sphere bundle, according to the value of this index. As applications, we obtain a complete classification of Zoll surfaces with boundary and of three-dimensional Zoll manifolds with boundary.

math.DG↗
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.

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↗
arXiv2024

Rotational Ricci surfaces

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature $K$ satisfies \begin{equation*} KΔK - \|\nabla K\|^2-4K^3 = 0. \end{equation*} These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces meeting the boundary of the unit Euclidean three-ball orthogonally. In addition, we show that this family interpolates a vertical geodesic and the critical catenoid.

math.DG↗
arXiv2022

A Note on Free Boundary Hypersurfaces in Space Forms Balls

In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we prove that the quantity of umbilical points of a free boundary surface in the unit ball counted with multiplicities depend only on its topology; moreover, we obtain as consequences that free boundary surfaces are annuli if, and only if, they have no umbilical points, and a new proof of the Nitsche Theorem. Secondly, we prove two geometric integral inequalities for free boundary hypersurfaces, and use them to relate some geometric aspects of the hypersurface with topological aspects of its boundary in the three-dimensional case, and to give a new point of view to the Catenoid Conjecture.

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↗