Colding–Minicozzi entropies in Cartan–Hadamard manifolds
-
Jacob Bernstein
and
Arunima Bhattacharya
Abstract
We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space. We show these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1904674
Award Identifier / Grant number: DMS-2203132
Funding statement: The first author was partially supported by the NSF Grants DMS-1904674 and DMS-2203132 and the Institute for Advanced Study with funding provided by the Charles Simonyi Endowment. The second author acknowledges the support of the AMS-Simons Travel Grant and funding provided by the Bill Guthridge Distinguished Professorship fund.
Acknowledgements
The authors thank Letian Chen and Junfu Yao for their helpful comments and suggestions. Part of the work was carried out when the second author was in residence at the Simons Laufer Mathematical Sciences Institute.
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- Colding–Minicozzi entropies in Cartan–Hadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical Lang–Siegel problem
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- Colding–Minicozzi entropies in Cartan–Hadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical Lang–Siegel problem
