Submanifolds immersed in a warped product with density

Araújo, Jogli G.; Lima, Henrique F. de; Gomes, Wallace F.; Velásquez, Marco Antonio L.

doi:10.36045/j.bbms.200126

Cited by 3 publications

(1 citation statement)

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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: 54%

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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