Some Sharp Isoperimetric Theorems for Riemannian Manifolds

By Frank Morgan, David L. Johnson. 2000

151 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/76aeacc2f00d8937b21d1544ccf94c253d5e6323

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds (2019, background): “The theorem, stated in [61, 60] for constant H , also holds for each H ≥ 0 non-increasing and is, in fact, a consequence of the concavity of the area of )Br as a function of the volume 7: s↦ vh ( V −1 h (s) ) is (strictly) concave on R.”
Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones (2003, methodology): “Similar approximation arguments were used by F. Morgan and D. Johnson [ MJ ] to obtain a differential inequality for the isoperimetric profile in a compact Riemannian manifold.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “We employ the following series of results in Riemannian Geometry, due to numerous groups of authors [11, 34, 73, 82, 51, 13, 12, 71, 16], who proved them under increasingly general conditions.”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “(1) It is well-known (see e.g. [6], [52], [7, Lemma 3.4.12]) that H μ ( A ) is essentially the derivative of I at v , or more precisely: (5.4) lim sup Recall also that by (3.3), int ( M ) ∩ A ⊂ ( − S ) + ∞ , where − S denotes the hypersurface S with reversed orientation (and hence mean-curvature).”
The isoperimetric profile of a smooth Riemannian manifold for small volumes (2007, unknown): “Remark: When we consider the solutions of the isoperimetric problem (this is the case treated in [ MJ00 ]), and not approximate solutions, the conclusion is stronger.”
Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound (2009, methodology): “In fact, by deriving a second-order differential inequality for I using the second variation, it is possible to show ([6],[44],[7],[8]) that under our smooth κ-semi-convexity assumptions, I is locally semi-concave, implying in particular that the right and left derivatives of I exist (and hence, coi”