Constant mean curvature graphs in a class of warped product spaces

2007 ◽  
Vol 131 (1) ◽  
pp. 173-179 ◽  
Author(s):  
Luis J. Alías ◽  
Marcos Dajczer
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Product Spaces ◽  
Warped Product Spaces

scholarly journals CONSTANT MEAN CURVATURE HYPERSURFACES IN WARPED PRODUCT SPACES

2007 ◽  
Vol 50 (3) ◽  
pp. 511-526 ◽  
Author(s):  
Luis J. Alías ◽  
Marcos Dajczer
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Complete Riemannian Manifold ◽  
General Situation ◽  
Warped Products ◽  
General Version ◽  
Product Spaces ◽  
Constant Mean Curvature Hypersurfaces ◽  
Warped Product Spaces

AbstractWe study hypersurfaces of constant mean curvature immersed into warped product spaces of the form $\mathbb{R}\times_\varrho\mathbb{P}^n$, where $\mathbb{P}^n$ is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces, which have recently been extensively studied. It also includes constant mean curvature hypersurfaces in the so-called pseudo-hyperbolic spaces. If the hypersurface is compact, we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\mathbb{R}\mapsto\{t\}\times\mathbb{P}^n$, generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.

scholarly journals A Sharp Height Estimate for Compact Hypersurfaces with Constant k-Mean Curvature in Warped Product Spaces

2014 ◽  
Vol 58 (2) ◽  
pp. 403-419 ◽  
Author(s):  
Sandra C. García-Martínez ◽  
Debora Impera ◽  
Marco Rigoli
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Higher Order ◽  
Product Spaces ◽  
Height Estimate ◽  
Higher Order Mean Curvature ◽  
Warped Product Spaces

AbstractIn this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher-order mean curvature and whose boundary is contained in a slice. We apply these results to draw topological conclusions at the end of the paper.

scholarly journals Warped product spaces with Ricci conditions

2017 ◽  
Vol 41 ◽  
pp. 1365-1375
Author(s):  
Sang Deok LEE ◽  
Byung Hak KIM ◽  
Jin Hyuk CHOI
Keyword(s):  
Warped Product ◽  
Product Spaces ◽  
Warped Product Spaces

scholarly journals Geodesic motion in the neighbourhood of submanifolds embedded in warped product spaces

2008 ◽  
Vol 40 (6) ◽  
pp. 1341-1351 ◽  
Author(s):  
Fábio Dahia ◽  
Carlos Romero ◽  
Lúcio F. P. da Silva ◽  
Reza Tavakol
Keyword(s):  
Warped Product ◽  
Geodesic Motion ◽  
Product Spaces ◽  
Warped Product Spaces

scholarly journals QUINTESSENTIAL INFLATION FROM 5D WARPED PRODUCT SPACES ON A DYNAMICAL FOLIATION

2008 ◽  
Vol 23 (16) ◽  
pp. 1213-1221 ◽  
Author(s):  
LUCIO FABIO P. DA SILVA ◽  
JOSÉ EDGAR MADRIZ AGUILAR
Keyword(s):  
Scalar Field ◽  
Equation Of State ◽  
Warped Product ◽  
Scalar Potential ◽  
Accelerated Expansion ◽  
Vacuum Equation ◽  
Product Spaces ◽  
Inflationary Scenario ◽  
Warped Product Spaces ◽  
The Universe

Assuming the existence of a 5D purely kinetic scalar field on the class of warped product spaces we investigate the possibility of mimic both an inflationary and a quintessential scenarios on 4D hypersurfaces, by implementing a dynamical foliation on the fifth coordinate instead of a constant one. We obtain that an induced chaotic inflationary scenario with a geometrically induced scalar potential and an induced quasi-vacuum equation of state on 4D dynamical hypersurfaces is possible. While on a constant foliation, the universe can be considered as matter-dominated today, in a family of 4D dynamical hypersurfaces, the universe can be passing period of accelerated expansion with a deceleration parameter nearly -1. This effect of the dynamical foliation results negligible at the inflationary epoch allowing for a chaotic inflationary scenario and becomes considerable at the present epoch allowing a quintessential scenario.

scholarly journals A note on Weingarten hypersurfaces in the warped product manifold

2014 ◽  
Vol 25 (14) ◽  
pp. 1450121 ◽  
Author(s):  
Haizhong Li ◽  
Yong Wei ◽  
Changwei Xiong
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Higher Order ◽  
Product Manifold ◽  
Weingarten Surfaces ◽  
Warped Product Manifold ◽  
Higher Order Mean Curvature ◽  
Special Case ◽  
Differential Geom

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

scholarly journals ON WARPED PRODUCT SPACES WITH A CERTAIN RICCI CONDITION

2013 ◽  
Vol 50 (5) ◽  
pp. 1683-1691 ◽  
Author(s):  
Byung Hak Kim ◽  
Sang Deok Lee ◽  
Jin Hyuk Choi ◽  
Young Ok Lee
Keyword(s):  
Warped Product ◽  
Product Spaces ◽  
Warped Product Spaces
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