We prove the existence of rotational hypersurfaces in H n × R with H r+1 = 0 and we classify them... more We prove the existence of rotational hypersurfaces in H n × R with H r+1 = 0 and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfaces.
We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclide... more We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclidean space Rm+n , m, n 3, analyzing also whether they are embedded or stable. We show also the existence of embedded, complete, stable minimal hypersurfaces in Rm+n , m + n 8, m, n 3 not homeomorphic to Rm+n−1 that are O(m) × O(n)-invariant. Mathematics Subject Classifications (2000): 53A10, 53C42.
We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for... more We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space.
We consider the integrals of r-mean curvatures Sr of a complete hypersurface M in space forms whi... more We consider the integrals of r-mean curvatures Sr of a complete hypersurface M in space forms which generalize volume (r = 0), total mean curvature (r = 1), total scalar curvature (r = 2) and total curvature (r = n). Among other results we prove that a complete properly immersed hypersurface of a space form with Sr ≥ 0, Sr ≡ 0 and S r+1 ≡ 0 for some r ≤ n -1 has R M SrdM = ∞.
We consider critical points of the functionals Π and Ψ defined as the global L 2 -norm of the sec... more We consider critical points of the functionals Π and Ψ defined as the global L 2 -norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion. We prove gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the sphere can be cast as a theorem about critical points of these functionals of constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various type of minimal submanifolds that can occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of Π.
We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-... more We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives
We prove that a H-surface M in H 2 × R, H ≤ 1 2 , inherits the symmetries of its boundary ∂M, whe... more We prove that a H-surface M in H 2 × R, H ≤ 1 2 , inherits the symmetries of its boundary ∂M, when ∂M is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.
We prove that a H-surface M in $${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$$ , in... more We prove that a H-surface M in $${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$$ , inherits the symmetries of its boundary $$\partial M,$$ when $$\partial M$$ is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic
In this work we consider connected, complete and orientable hypersurfaces of the sphere S n+1 wit... more In this work we consider connected, complete and orientable hypersurfaces of the sphere S n+1 with constant nonnegative r-mean curvature. We prove that under subsidiary conditions, if the Gauss image of M is contained in a closed hemisphere, then M is totally umbilic.
We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+... more We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfaces.
In this survey paper, we review several results connecting geometric properties(compactness, fini... more In this survey paper, we review several results connecting geometric properties(compactness, finiteness of the Morse index, caracterization of stablesurfaces) and curvature estimates for constant mean curvature submanifoldsin simply-connected space forms. The results go in both directions(curvature estimates to geometric properties and conversely).Neste artigo descreveremos v'arios resultados relacionando propriedadesgeom'etricas (compacidade, "indice de Morse, caracteriza?c~ao de...
ABSTRACT The main result of this paper states that the traceless second fundamental tensor A0 of ... more ABSTRACT The main result of this paper states that the traceless second fundamental tensor A0 of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, [`(M)]\bar M (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H 2 + c > 0, any such surface must be compact.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegativ... more We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero Gauss-Kronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).
We prove the existence of rotational hypersurfaces in H n × R with H r+1 = 0 and we classify them... more We prove the existence of rotational hypersurfaces in H n × R with H r+1 = 0 and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfaces.
We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclide... more We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclidean space Rm+n , m, n 3, analyzing also whether they are embedded or stable. We show also the existence of embedded, complete, stable minimal hypersurfaces in Rm+n , m + n 8, m, n 3 not homeomorphic to Rm+n−1 that are O(m) × O(n)-invariant. Mathematics Subject Classifications (2000): 53A10, 53C42.
We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for... more We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space.
We consider the integrals of r-mean curvatures Sr of a complete hypersurface M in space forms whi... more We consider the integrals of r-mean curvatures Sr of a complete hypersurface M in space forms which generalize volume (r = 0), total mean curvature (r = 1), total scalar curvature (r = 2) and total curvature (r = n). Among other results we prove that a complete properly immersed hypersurface of a space form with Sr ≥ 0, Sr ≡ 0 and S r+1 ≡ 0 for some r ≤ n -1 has R M SrdM = ∞.
We consider critical points of the functionals Π and Ψ defined as the global L 2 -norm of the sec... more We consider critical points of the functionals Π and Ψ defined as the global L 2 -norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion. We prove gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the sphere can be cast as a theorem about critical points of these functionals of constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various type of minimal submanifolds that can occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of Π.
We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-... more We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives
We prove that a H-surface M in H 2 × R, H ≤ 1 2 , inherits the symmetries of its boundary ∂M, whe... more We prove that a H-surface M in H 2 × R, H ≤ 1 2 , inherits the symmetries of its boundary ∂M, when ∂M is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.
We prove that a H-surface M in $${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$$ , in... more We prove that a H-surface M in $${\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2$$ , inherits the symmetries of its boundary $$\partial M,$$ when $$\partial M$$ is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic
In this work we consider connected, complete and orientable hypersurfaces of the sphere S n+1 wit... more In this work we consider connected, complete and orientable hypersurfaces of the sphere S n+1 with constant nonnegative r-mean curvature. We prove that under subsidiary conditions, if the Gauss image of M is contained in a closed hemisphere, then M is totally umbilic.
We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+... more We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfaces.
In this survey paper, we review several results connecting geometric properties(compactness, fini... more In this survey paper, we review several results connecting geometric properties(compactness, finiteness of the Morse index, caracterization of stablesurfaces) and curvature estimates for constant mean curvature submanifoldsin simply-connected space forms. The results go in both directions(curvature estimates to geometric properties and conversely).Neste artigo descreveremos v'arios resultados relacionando propriedadesgeom'etricas (compacidade, "indice de Morse, caracteriza?c~ao de...
ABSTRACT The main result of this paper states that the traceless second fundamental tensor A0 of ... more ABSTRACT The main result of this paper states that the traceless second fundamental tensor A0 of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, [`(M)]\bar M (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H 2 + c &gt; 0, any such surface must be compact.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegativ... more We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero Gauss-Kronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).
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