https://neckpinch.com/authors/jacob-bernstein/New arXiv papers by Jacob Bernstein, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2407.02332arxiv:2407.02332Tue, 02 Jul 2024 00:00:00 GMTJacob Bernstein — We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest.Mean curvature flowSelf-shrinkers & translatorsHarnack inequalitiesSurfaces https://arxiv.org/abs/2306.11618arxiv:2306.11618Tue, 20 Jun 2023 00:00:00 GMTJacob Bernstein — We show certain rigidity for minimizers of generalized Colding-Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.Mean curvature flowStability & uniqueness https://arxiv.org/abs/2211.14257arxiv:2211.14257Fri, 25 Nov 2022 00:00:00 GMTJacob Bernstein, Arunima Bhattacharya — We introduce a family of functionals on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. We show that 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.Mean curvature flowStability & uniqueness