A note on the characterization of spheres as self-shrinkers
- PDF / 242,051 Bytes
- 3 Pages / 439.37 x 666.142 pts Page_size
- 113 Downloads / 209 Views
Archiv der Mathematik
A note on the characterization of spheres as self-shrinkers Wagner O. Costa-Filho
Abstract. We characterize spheres as the unique complete properly immersed self-shrinkers in arbitrary codimension satisfying a geometric inequality. Mathematics Subject Classification. 53C42, 53C44. Keywords. Self-shrinkers, Immersed submanifolds, Minkowski formula.
1. Introduction. Let x : Σn → Rn+p be a complete oriented submanifold immersed in the Euclidean space. Let us denote by II the second fundamental form and by H its (non normalized) mean curvature vector field, that is H = tr(II). We say that Σ is a self-shrinker of the mean curvature flow if it satisfies x⊥ , (1.1) 2 where ⊥ denotes the projection onto the normal bundle of Σ (see for instance [1,2]). Self-shrinkers form an important class of submanifolds for both the geometrical and the analytical point of view and appear naturally in many different contexts. In this note, we present a simple proof of the following result, originally proved by Vasquez [4] in the case of embedded compact self-shrinkers of codimension one. We point out that the constant in our definition (1.1) is given such that it coincides with the definition presented in [4] when √ p = 1. Moreover, the round sphere Sn (r) is a self-shrinker if and only if r = 2n. H=−
Theorem 1.1. Let Σ be a complete properly immersed self-shrinker in Rn+p . If n+2 2 2 1− |x| + 2 |H|2 − |II|2 ≥ 0 (1.2) 4n2 n √ on Σ, then Σ is a round sphere of radius r = 2n.
W.O. Costa-Filho
Arch. Math.
Proof. We first recall that the traceless second fundamental form is the tensor defined by 1 Φ(X, Y ) = II(X, Y ) − X, Y H. n So, |Φ|2 = |II|2 − n1 |H|2 and |Φ| ≡ 0 if and only if the immersion is totally umbilical. The self-shrinker equation (1.1) implies that 4|H|2 = |x⊥ |2 ≤ |x|2 . So, from our hypothesis (1.2), we obtain n+2 2 1 ⊥2 |x | ≤ 1, |x | + (1.3) 2 4n 4n and (1.4) n|II|2 ≤ n − |H|2 . We claim that Σ is compact. In fact, from inequality (1.3), Σ is bounded. Since Σ is properly immersed, it is closed in Rn+p , and the claim follows. Now, integrating inequality (1.4) and rearranging its terms, we get 2 2 2 0 ≤ n|Φ| dΣ = n|II| − |H| dΣ ≤ n − 2|H|2 dΣ. Σ
Σ
Σ
We recall the Minkowski formula for immersed submanifolds as proved by Str¨ ubing [3]: 0 = (n + x, H) dΣ. Σ
Thus, using the self-shrinker equation, we obtain 0= n − 2 |H|2 dΣ. Σ 2
It implies that |Φ| ≡ 0 and thus Σ is totally umbilical.
Acknowledgements. The author is deeply grateful to Professor Marcos P. Cavalcante for his continued guidance and support in preparing this paper, and to Professor Detang Zhou for enlightening comments. He also thanks the referee for valuables comments and for pointing out that inequality (1.2) implies compactness. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References [1] Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879–889 (
Data Loading...
