Colding Minicozzi entropy in hyperbolic space

Superconvexity of the Heat Kernel on Hyperbolic Space

“…In [1], Bernstein proved (1.2) for small n and conjectured it for all n. Hence we confirm this conjecture in Theorem 1.1.…”

“…By the symmetries of H n , there is a positive function K n (t, ρ) on (0, ∞) × (0, ∞) such that H n (t, p; t 0 , p 0 ) = K n (t 0 − t, dist H n (p, p 0 )) > 0 where ρ = dist H n (p, p 0 ) is the hyperbolic distance between p and p 0 . As remarked in [1], although K n can be explicitly computed, the formulas become unmanageable for large n; see [8] for more details.…”

“…Remark 1.3. In [1], Bernstein introduced a notion of hyperbolic entropy for submanifolds in hyperbolic space, which is analogous to the one introduced by Colding-Minicozzi for hypersurfaces in Euclidean space [7]. Using Corollary 1.2 and some observations of [1], one may adapt the arguments of [2-4, 6, 9, 12] to prove that closed hypersurfaces in hyperbolic space with small hyperbolic entropy are simple in various senses.…”