Stable and isoperimetric regions in some weighted manifolds with\n boundary

2020/07/26 by César Rosales, Rosales, César
Mathematics · Engineering · #Geometric Analysis and Curvature Flows #Elasticity and Material Modeling #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.13107

Abstract

In a Riemannian manifold with a smooth positive function that weights the\nassociated Hausdorff measures we study stable sets, i.e., second order minima\nof the weighted perimeter under variations preserving the weighted volume. By\nassuming local convexity of the boundary and certain behaviour of the\nBakry- 'Emery-Ricci tensor we deduce rigidity properties for stable sets by\nusing deformations constructed from parallel vector fields tangent to the\nboundary. As a consequence, we completely classify the stable sets in some\nRiemannian cylinders \Ω\×\ℝ with product weights. Finally, we\nalso establish uniqueness results showing that any minimizer of the weighted\nperimeter for fixed weighted volume is bounded by a horizontal slice\n\Ω\× t .\n

Related

• Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound
• Isoperimetric and stable sets for log-concave perturbations of Gaussian measures
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• Isoperimetric inequalities in cylinders with density
• Rigidity of manifolds admitting stable solutions of an elliptic problem