A tangency principle and applications

Fontenele, Francisco; Silva, Sérgio L.

doi:10.1215/ijm/1258138264

Cited by 24 publications

(32 citation statements)

References 8 publications

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“…The following theorem extends for Euclidean hypersurfaces with the Jordan-Brouwer property results of Blaschke [2], Koutroufiotis [9] and the authors [5] for compact Euclidean hypersurfaces. THEOREM 1.3.…”

Section: Introductionmentioning

confidence: 58%

“…As in [5], given p ∈ M, we can parametrize a neighbourhood of M n containing p and contained in a normal ball of N n+1 putting…”

Section: A Tangency Principlementioning

confidence: 99%

“…From now on |σ 1 |(x) and |σ 2 |(x) will represent the length of the second fundamental form of M n 1 at ϕ 1 (x) and M n 2 at ϕ 2 (x) respectively. As in [5], we say that M n 1 remains above M n 2 in a neighbourhood of p with respect to η o when µ 1 (x) µ 2 (x) in a neighbourhood of zero in T p M 1 = T p M 2 . Following the ideas in [5], we obtain the following tangency principle: For hypersurfaces with boundaries and as a consequence of proof of Theorem 2.1, we obtain the following tangency principle: Theorem 2.3 is a particular case of a result due to Alexandrov (see [1], Theorem 5 on page 347 and Section 4).…”

Section: A Tangency Principlementioning

confidence: 99%

“…In [5], the same authors proved that if an n-dimensional compact Riemannian manifold M n embedded into the (n + 1)-dimensional Euclidean space satisfies |H k | 1 λ k , for some k-mean curvature H k , 1 k n, and some positive constant λ, then the greatest ball that fits inside M n has radius less than λ unless M n be a sphere of radius λ, extending results of Blaschke and Koutroufiotis for surfaces (see [2,9]). In this paper we improve (see Remark 5.1) this radius estimate and extend it to Euclidean hypersurfaces with the Jordan-Brouwer property.…”

Section: Introductionmentioning

confidence: 99%

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Sharp Estimates for the Size of Balls in the Complement of a Hypersurface

Fontenele

Silva

2005

Geom Dedicata

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In this paper, we make estimates for the radius of balls contained in some component of the complementary of a complete hypersurface into a space form, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The results are obtained using an algebraic lemma and a tangency principle related with the length of the second fundamental form. The algebraic lemma also is used to improve a result for graphs due to Hasanis-Vlachos. Mathematics Subject Classifications (2000). 53C42, 53C21, 35B50, 35J60.

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Section: Introductionmentioning

confidence: 58%

“…As in [5], given p ∈ M, we can parametrize a neighbourhood of M n containing p and contained in a normal ball of N n+1 putting…”

Section: A Tangency Principlementioning

confidence: 99%

Section: A Tangency Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sharp Estimates for the Size of Balls in the Complement of a Hypersurface

Fontenele

Silva

2005

Geom Dedicata

Self Cite

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“…The surface Σ 2 ⊂ H 3 parameterized by φ(u, v) = (u, v, e v ) (considering H 3 in the Poincaré half-space model) is properly embedded with a single point in its asymptotic boundary and mean curvature satisfying |H| = (2+3e v )/(2(1 + e 2v ) 3 2 ) < 1, which shows that the sup-norm in Theorem 1.2 is essential. This example was exhibited by Lluch [7].…”

Section: Introductionmentioning

confidence: 93%

The influence of the boundary behavior on isometric immersions in the hyperbolic space

2009

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This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface f : Σ n → H n+1 with mean curvature satisfying sup p∈Σ H(p) < 1 has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. Mathematics Subject Classification (2000). Primary 53C42; Secondary 53C40.

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