Optimal constants of $$L^2$$ L 2 inequalities for closed nearly umbilical hypersurfaces in space forms

Cheng, Xu; Juárez, Areli Vázquez

doi:10.1007/s10711-014-9985-z

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…The W 2,2 -closeness result was reproved with the help of the Willmore flow in [KS20]. The version for hyperbolic and spherical spaces was treated in [CZ14] and some optimality results in [CJ15]. Further results of almost umbilical type in terms of other L p -norms were deduced in [Per11,Rot13,Rot15,RS18,Sch15].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Stability from rigidity via umbilicity

Scheuer¹

2021

Preprint

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We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the Hausdorff distance to a geodesic sphere of an embedded hypersurface to the L 2 -norm of the traceless Hessian operator of a defining function for the open set bounded by the hypersurface. As application, this result allows for a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff's soap bubble theorem in space forms, Serrin's overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff-Fenchel inequalities.

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Section: Introduction and Resultsmentioning

confidence: 99%

Stability from rigidity via umbilicity

Scheuer¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Stability from rigidity via umbilicity

Scheuer

2024

Advances in Calculus of Variations

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We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff’s soap bubble theorem in space forms, Serrin’s overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff–Fenchel inequalities.

show abstract

Optimal constants of $$L^2$$ L 2 inequalities for closed nearly umbilical hypersurfaces in space forms

Cited by 2 publications

References 17 publications

Stability from rigidity via umbilicity

Stability from rigidity via umbilicity

Stability from rigidity via umbilicity

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