Nonlinear evolution by mean curvature and isoperimetric inequalities

By F. Schulze. 2006

90 citations recorded by the source corpus.

Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n <= 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9ea475a20693cbf3937171d7c1448a25018ca6a1

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, methodology): “Various flows have been employed to prove geometric inequalities, see for instance [2, 5, 13, 18, 19, 21, 24, 25, 26, 29, 30, 31, 38].”
Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature (2018, background): “Acknowledgements The authors are grateful to C. Arezzo, A. Carlotto, A. Farina, G. Huisken, L.Mari, D. Peralta-Salas, F. Schulze and P. Topping for useful comments and discussions during the preparation of the paper.”
A Mean Curvature Type Flow in Space Forms (2013, methodology): “Schulze [23] used a non-linear mean curvature type flow to prove the isoperimetric inequality in Rn+1 with n ≤ 7.”