Papers by Feliciano Vitório
arXiv (Cornell University), Oct 5, 2012
In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in q... more In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanuša, Rozendorn, Henke and Azov. 2010 Mathematics Subject Classification. Primary 53C20; Secondary 31C05. The authors were partially supported by CNPq-Brazil. 1 We recall that an injective immersion is an embedding if it is a homeomorphism onto its image, by considering the image with the induced topology.
arXiv (Cornell University), Feb 12, 2015
In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds w... more In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds with warped structure.
arXiv (Cornell University), May 24, 2021
In this article, we establish a relationship between geometric quantities of a hypersurface restr... more 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.
arXiv (Cornell University), Nov 28, 2019
In this article, we study constant mean curvature isometric immersions into S 2 × R and H 2 × R a... more In this article, we study constant mean curvature isometric immersions into S 2 × R and H 2 × R and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the 3−dimensional homogenous manifolds E(κ, τ) and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in S 2 × S 2 and H 2 × H 2 with constant intrinsic curvature. It is worthwhile to point out that these classifications provide new examples.
Journal of Mathematical Analysis and Applications, Sep 1, 2023
In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immerse... more In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.
Annals of Global Analysis and Geometry, Apr 30, 2019
In this paper, we prove that there exists a universal constant C, depending only on positive inte... more In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .
arXiv (Cornell University), Jul 18, 2018
In this paper, we prove that there exists a universal constant C, depending only on positive inte... more In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .
arXiv (Cornell University), Oct 2, 2018
Let M be a compact connected surface with boundary. We prove that the signal condition given by t... more Let M be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function f on ∂M (resp. on M) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on M (resp. metric on M with geodesic boundary). In order to provide analogous results for this problem with n ≥ 3, we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on ∂M (resp. on M) is a mean curvature of a scalar flat metric on M (resp. scalar curvature of a metric on M and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.
Annales de l'Institut Fourier, May 12, 2023
Article à paraître Mis en ligne le 29 juillet 2022.
arXiv (Cornell University), Apr 7, 2022
We consider the problem of studying the set of curvature functions which a given compact and non-... more We consider the problem of studying the set of curvature functions which a given compact and non-compact manifold with nonempty boundary can possess. First we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our proof conceptually differs from [17] since our approach allow us to solve problems where the conformal method cannot solve. Also, we prove new existence and nonexistence of metrics with prescribed curvature in the conformal setting which depends on the Euler characteristic. After this, we present a higher order analogue concerning scalar and mean curvatures on compact manifolds with boundary. We also give conditions for Riemannian manifolds not necessarily complete or compact to admit positive scalar curvature and minimal boundary, without any auxiliary assumptions about its "infinity", which is an extension of those proved by Carlotto-Li [11].
Journal of Mathematical Analysis and Applications
In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immerse... more In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.
arXiv (Cornell University), Feb 16, 2023
Annales de l'Institut Fourier
Article à paraître Mis en ligne le 29 juillet 2022.
arXiv: Differential Geometry, 2015
After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sob... more After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the dimension of the submanifold. A sort of applications to the submanifold theory and geometric analysis have been obtained from that inequality. It is worthwhile to point out that, by a Nash Theorem, every Riemannian manifold can be seen as a submanifold in some Euclidean space. In the same spirit, Carron obtained a Hardy inequality for submanifolds in Euclidean spaces. In this paper, we will prove the Hardy, weighted Sobolev and Caffarelli-Kohn-Nirenberg inequalities, as well as some of their derivatives, as Galiardo-Nirenberg and Heisenberg-Pauli-Weyl inequalities, for submanifolds in a class of manifolds, that include, the Cartan-Hadamard ones.
Annals of Global Analysis and Geometry, 2019
In this paper, we prove that there exists a universal constant C, depending only on positive inte... more In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .
Journal of Mathematical Analysis and Applications, 2016
Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of me... more Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is need.
Journal of Differential Equations, 2017
Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in E... more Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon [10] and Hoffman-Spruck [9]. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. Years later, Carron [4] obtained a Hardy inequality for submanifolds in Hadamard spaces. In this paper, we prove the general Hardy and Rellich Inequalities for submanifolds in Hadamard spaces. Some applications are given and we also analyse the equality cases.
Results in Mathematics, 2016
In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds w... more In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds with warped structure.
Kodai Mathematical Journal, 2015
In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in q... more In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanuša, Rozendorn, Henke and Azov.
Proceedings of the American Mathematical Society, 2014
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Papers by Feliciano Vitório