Self-shrinkers with bounded HA

Abstract

We study integral and pointwise bounds on the second fundamental form of properly immersed self-shrinkers with bounded HA. As applications, we discuss gap and compactness results for self-shrinkers. Contents 1 Introduction 2 Preliminaries 3 L p estimate and growth rate 4 Gap and compactness theorems From [3] one sees the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space. Lemma 2.3. (Theorem 1.1 of [3]) Let Σ n be a complete noncompact properly immersed self-shrinker in Eucildean space R n+1 . Then Σ has finite weighted volume Volwhere C is a positive constant depending only on Σ e −f dv.Provided bounded mean curvature we can also get volume ratio lower bound. See Lemma 3.5 in Li-Wang [15].Lemma 2.4. (Lemma 3.5 of [15]) Let Σ n ֒→ R n+1 be a properly immersed hypersurface in B(x 0 , r 0 ) with x 0 ∈ Σ and sup Σ |H| ≤ Λ. Then for any s ∈ (0, r 0 ) we have Vol Σ (B(x 0 , s) ∩ Σ)ω n s n ≤ e Λr 0 Vol Σ (B(x 0 , r 0 ) ∩ Σ) ω n r 0 n .In particular, Vol(B(x 0 , r) ∩ Σ) ≥ e −Λr ω n r n , ∀ r ∈ (0, r 0 ].