Sobolev and isoperimetric inequalities for submanifolds in weighted ambient spaces

Batista, Márcio; Mirandola, Heudson

doi:10.1007/s10231-014-0449-8

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…for all ξ ∈ Σ, where ω n denotes the volume of n-dimensional unit ball in R n . We remark that Batista-Mirandola [3] also obtained a similar Sobolev inequality in the weighted setting independently. The proof of Theorem 4.1 in [12] used the heat kernel and Michael-Simon Sobolev inequality, inspired by a similar argument in Li-Yau's paper [30].…”

Section: Estimate On the Upper Bound Of The F -Indexmentioning

confidence: 53%

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Barbosa

Sharp

Wei

2017

Proc. Amer. Math. Soc.

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Abstract. Let (M n+1 , g, e −f dµ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded f -minimal hypersurfaces in M with uniform upper bounds on f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n+1 with 2 ≤ n ≤ 6. We also prove some estimates on the f -index of f -minimal hypersurfaces, and give a conformal structure of f -minimal surface with finite f -index in three-dimensional smooth metric measure space.

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Section: Estimate On the Upper Bound Of The F -Indexmentioning

confidence: 53%

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Barbosa

Sharp

Wei

2017

Proc. Amer. Math. Soc.

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“…Bellow, let us denote by H the mean curvature vector field of an isometric immersion x : M m →M and by ||H|| L q (E) its Lebesgue L q -norm on E ⊂ M . As a direct consequence, we have: The main tool in the proof of Theorem 1.2 is the Hofmann-Spruck inequality [5] and its refinement given in [2].…”

Section: Introductionmentioning

confidence: 90%

The $$p$$ p -hyperbolicity of infinity volume ends and applications

2013

Self Cite

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“…The Hoffman-Spruck's Theorem above can be generalized for ambient spaces M satisfying ( Krad ) ξ ≤ K(r ξ ), for all ξ ∈ M . The details and proof for this case can be found, for instance, in [2].…”

Section: The Weighted Hoffman-spruck Inequality For Submanifoldsmentioning

confidence: 99%

The Caffarelli-Kohn-Nirenberg Inequality for Submanifolds in Riemannian Manifolds

Batista,

Mirandola,

Vitório

2015

Preprint

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After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the dimension of the submanifold. A sort of applications to the submanifold theory and geometric analysis have been obtained from that inequality. It is worthwhile to point out that, by a Nash Theorem, every Riemannian manifold can be seen as a submanifold in some Euclidean space. In the same spirit, Carron obtained a Hardy inequality for submanifolds in Euclidean spaces. In this paper, we will prove the Hardy, weighted Sobolev and Caffarelli-Kohn-Nirenberg inequalities, as well as some of their derivatives, as Galiardo-Nirenberg and Heisenberg-Pauli-Weyl inequalities, for submanifolds in a class of manifolds, that include, the Cartan-Hadamard ones.

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Sobolev and isoperimetric inequalities for submanifolds in weighted ambient spaces

Abstract: In this paper, we prove Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities (Hoffman and Spruck in Commun Pure Appl Math 27:715-727, 1974).

Cited by 9 publications

References 18 publications

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

The $$p$$ p -hyperbolicity of infinity volume ends and applications

The Caffarelli-Kohn-Nirenberg Inequality for Submanifolds in Riemannian Manifolds

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