Existence of integral $$m$$ -varifolds minimizing $$\int \!|A|^p$$ and $$\int \!|H|^p,\,p&gt;m,$$ in Riemannian manifolds

Mondino, Andrea

doi:10.1007/s00526-012-0588-y

Cited by 13 publications

(4 citation statements)

References 27 publications

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“…The proof of the following lemma is based on the ideas of the monotonicity formula in Simon [39] in combination with a technique of Anderson [3]. See also [31,Lemma A.3] for a proof in the presence of boundary, and [25] for higher-dimensional varifolds.…”

Section: Monotonicity Inequalitiesmentioning

confidence: 99%

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Scharrer

2022

Ann Glob Anal Geom

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Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.

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Section: Monotonicity Inequalitiesmentioning

confidence: 99%

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Scharrer

2022

Ann Glob Anal Geom

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show abstract

“…The proof of the following Lemma is based on the ideas of the monotonicity formula in Simon [38] in combination with a technique of Anderson [3]. See also [30,Lemma A.3] for a proof in the presence of boundary, and [25] for higher dimensional varifolds.…”

Section: Monotonicity Inequalitiesmentioning

confidence: 99%

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Scharrer

2021

Preprint

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Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds.

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“…The analysis of the long time behaviour of ∇E−flows discussed in the present work can be adapted also to the case of the Willmore flow of surfaces into Riemannian manifolds. In particular, under conditions ensuring the uniform boundedness of the area of the evolving surfaces (see [12] for an analysis of some special cases) and guaranteeing the existence of immersions F : Σ → N 3 with W(F ) < 4π, the analogues of Theorem 6.3 and Proposition 6.7 can be proven.…”

Section: Long Time Existencementioning

confidence: 99%

A convergence result for the Gradient Flow of ∫ |A|2 in Riemannian Manifolds

Magni¹

2015

Geometric Flows

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We study the gradient flow of the L 2 −norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L 2 −concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.

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Existence of integral $$m$$ -varifolds minimizing $$\int \!|A|^p$$ and $$\int \!|H|^p,\,p>m,$$ in Riemannian manifolds

Cited by 13 publications

References 27 publications

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

A convergence result for the Gradient Flow of ∫ |A|2 in Riemannian Manifolds

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