Maximum principles for hypersurfaces with vanishing curvature functions in an arbitrary Riemannian manifold

Fontenele, Francisco; Silva, Sérgio L.

doi:10.1590/s0001-37652002000200002

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Maximum Principle and Asymptotic Theorems2

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2018

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“…M aximum principle [, Theorem 2.a]. Let

M

and

M^{'}

two oriented hypersurfaces with

H_{r} = H_{r}^{'} \equiv 0

, tangent at a point

p

, with normal vector pointing in the same direction.…”

Section: Maximum Principle and Asymptotic Theoremsmentioning

confidence: 99%

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

Elbert¹,

Nelli

2018

Bull. London Math. Soc.

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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

“…M aximum principle [, Theorem 2.a]. Let

M

and

M^{'}

two oriented hypersurfaces with

H_{r} = H_{r}^{'} \equiv 0

, tangent at a point

p

, with normal vector pointing in the same direction.…”

Section: Maximum Principle and Asymptotic Theoremsmentioning

confidence: 99%

“…The suitable version of maximum principle for our purposes is stated below. For further details about such generalized maximum principles, see [7,8,11,12]. Theorem 4.1.…”

Section: Maximum Principle and Asymptotic Theoremsmentioning

confidence: 99%