On Submanifolds With Tamed Second Fundamental Form

Bessa, G. Pacelli; Costa, M. S.

doi:10.1017/s0017089509990085

Cited by 9 publications

(31 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result, due to Gimeno-Palmer [13], extends Bessa-Montenegro-Jorge [5] and Bessa-Costa [4]. We shall present our proof of theorem 2.2 for the sake of completeness and to clarify the notation used in the paper.…”

Section: Preliminariesmentioning

confidence: 68%

“…they are properly immersed and have finite topology, meaning that M is C ∞ -diffeomorphic to a compact smooth manifold M with boundary. This result was extend by Bessa-Costa to isometric immersions with tamed second fundamental form into Hadamard manifolds, [4] and by Gimeno-Palmer in [13] to isometric immersion with tamed second fundamental form into ambient manifolds with a pole and bounded radial sectional curvatures. They also have shown that the volume growth and the number of ends of submanifolds of dimension greater than 2 are controlled with an appropriate decay of the extrinsic curvature.…”

Section: Introductionmentioning

confidence: 70%

“…where ρ M (x) = dist M (x 0 , x) is the distance function on M to a fixed point x 0 , and α(x) is the norm of the second fundamental form at ϕ(x). The notion of immersion with tamed second fundamental form was introduced in [5] for submanifolds of R n and in [4] for submanifolds of Hadamard manifolds. This notion can be naturally extended to manifolds with a pole and radial sectional curvature bounded above, see [13].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

Brandao¹,

Gimeno

2016

Differential Geometry and its Applications

View full text Add to dashboard Cite

ABSTRACT. In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In the particular case of a surface of nonnegative curvature, we prove that the surface is diffeomorphic to the Euclidean plane if the surface has tamed second fundamental form, and that the surface is isometric to the Euclidean plane if the surface has strongly tamed second fundamental form. In the last part of the paper we characterize the fundamental tone of any submanifold of tamed second fundamental form immersed in an ambient space with a pole and quadratic decay of the radial sectional curvatures.

show abstract

Section: Preliminariesmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

Brandao¹,

Gimeno

2016

Differential Geometry and its Applications

View full text Add to dashboard Cite

show abstract

“…This is a contradiction and proves that

M

is properly immersed. Now we use that

H^{n} \times R

is a Hadamard manifold and we note that Proposition implies that

X : M^{n} \to H^{n} \times R

has tamed second fundamental form (see [, Definition 1.1]). Then can use (the proof of) [, Theorem 1.2] to conclude there exists a ball of

H^{n} \times R

, centred at the origin, of radius

r_{0}

such that the extrinsic distance has no critical points outside this ball.

□

…”

Section: Finite Strong Total Curvaturementioning

confidence: 99%

“…Remark The technique of the proof of [, Theorem 1.2] can also be used to get an alternative proof of the fact that

X

is properly immersed.…”

Section: Finite Strong Total Curvaturementioning

confidence: 99%

On the structure of hypersurfaces in Hn×R with finite strong total curvature

Elbert¹,

Nelli

2018

Bull. London Math. Soc.

View full text Add to dashboard Cite

We prove that if X:Mn→Hn×R, n⩾3, is an orientable, complete immersion with finite strong total curvature, then X is proper and M is diffeomorphic to a compact manifold trueM¯ minus a finite number of points q1,⋯qk. Adding some extra hypothesis, including Hr=0, where Hr is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of Hn×R which satisfy Hr=0 and are invariant by hyperbolic translations and a maximum principle in a half‐space for these hypersurfaces.

show abstract

Density and spectrum of minimal submanifolds in space forms

Lima

Mari²,

Montenegro³

et al. 2016

Math. Ann.

View full text Add to dashboard Cite

Abstract. Let ϕ : M m → N n be a minimal, proper immersion in an ambient space suitably close to a space form N n k of curvature −k ≤ 0. In this paper, we are interested in the relation between the density function Θ(r) of M and the spectrum of its Laplace-Beltrami operator. In particular, we prove that if Θ(r) has subexponential growth (when k < 0) or sub-polynomial growth (k = 0) along a sequence, then the spectrum of M m is the same as that of the space form N m k . Notably, the result applies to Anderson's (smooth) solutions of Plateau's problem at infinity on the hyperbolic space, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures M to have finite density. In particular, we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density. density and spectrum and Laplace-Beltrami and minimal submanifolds and monotonicity

show abstract

On Submanifolds With Tamed Second Fundamental Form

Cited by 9 publications

References 14 publications

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

On the structure of hypersurfaces in Hn×R with finite strong total curvature

Density and spectrum of minimal submanifolds in space forms

Contact Info

Product

Resources

About