Riemannian Submanifolds: A Survey

Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow

“…Remark o. It is easy to check that the bound for the squared norm of the second fundamental form 2 3 n, used in [20] and which do not depends on the codimension m, is bigger or equal than the bound m−1 2m−3 n used in [35], [9], [2], when m ≥ 3. In fact, for all n > 0, the values are equal when m = 3 and 2 3 n > m−1 2m−3 n when m > 3.…”

“…To do that, let us consider the function r 2 : Σ → R, defined as r 2 (p) = X(p) 2 , where r is the extrinsic distance to 0 in Σ ⊆ R n+m . Then, applying Lemma 2.2 to the radial function F (r) = r 2 (7.8)…”

“…Let X : Σ n → R n+m be a complete properly immersed λ-self-shrinker in R n+m for the Mean Curvature Flow, (MCF), with respect 0 ∈ R n+m . Let us suppose that: 2) or Σ yields entirely out this ball,…”

“…To prove Theorem 7.10 we will make use of the classification provided by J. Simon, and S.S. Chern, M.P. Do Carmo and S. Kobayashi, for compact minimal immersions in the sphere, (see [35], [9], [2]), refined later by A.M. Li and J.M. Li, ( see [20]).…”