Two-sided hypersurfaces, entire Killing graphs and the mean curvature equation in warped products with density

Lima, Henrique F. de; Ramalho, André F. A.; Velásquez, Marco Antonio L.

doi:10.1016/j.difgeo.2020.101623

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…As a consequence, we deduce half-space and Bernstein type theorems for hypersurfaces with non-empty boundary in some weighted manifolds. Related results for hypersurfaces with empty boundary have been established by many authors in several previous works, see for instance [41,32,20,38,10,9,30,8,7,21,16,14,40,13,1,15,2,19,27,33]. We also generalize to the weighted context some well-known enclosure properties for compact minimal surfaces with boundary in R 3 , like the convex hull property, the hyperboloid theorem and the cone theorem, see [17,Ch.…”

Section: Introductionmentioning

confidence: 55%

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Castro¹,

Rosales²

2022

Preprint

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For a Riemannian manifold M , possibly with boundary, we consider the Riemannian product M × R k with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with non-empty boundary and contained within certain regions of M × R k with suitable weights. Our results include half-space and Bernsteintype theorems in weighted cylinders. We also generalize to this setting some classical properties about the confinement of a compact minimal hypersurface to certain regions of Euclidean space according to the position of its boundary. Finally, we show interesting situations where the statements are applied, some of them in relation to the singularities of the mean curvature flow.

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Section: Introductionmentioning

confidence: 55%

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Castro¹,

Rosales²

2022

Preprint

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“…Moreover, [5] established Liouville-type results related to twosided hypersurfaces immersed in a weighted Killing warped product. More recently, some uniqueness results of complete two-sided hypersurfaces in warped products with density are given in [6].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

Zhang

2021

Advances in Mathematical Physics

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In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρ M f n whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.

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Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Castro

Rosales

2022

Journal of Mathematical Analysis and Applications

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Two-sided hypersurfaces, entire Killing graphs and the mean curvature equation in warped products with density

Cited by 3 publications

References 23 publications

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

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