arXiv:2106.09165v1 [math.DG] 16 Jun 2021 STABILITY PROPERTIES OF COMPLETE SELF-SHRINKING SURFACES IN R 3 HIL ´ ARIO ALENCAR, GREG ´ ORIO SILVA NETO & DETANG ZHOU Abstract. This paper studies rigidity for immersed self-shrinkers of the mean curva- ture flow of surfaces in the three-dimensional Euclidean space R 3 . We prove that an immersed self-shrinker with finite L-index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in R 3 without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza. 1. Introduction A n-dimensional self-shrinker of the mean curvature flow in R n+1 is a hypersurface Σ which satisfies H (x)= 1 2 〈x, ν (x)〉, where H (x) is the mean curvature of Σ at x ∈ Σ and ν is its outward unitary normal vector field. Here we are using the convention of [8] such that the mean curvature of a n-dimensional round sphere of radius R is n/R, and H = trace A, where A(Y )= ∇ Y ν, for Y ∈ T Σ, and ∇ is the connection of R n+1 . Self-shrinkers are known as type I singularities of the mean curvature flow. They can also be seen as the critical points of the weighted area functional ˆ Σ e − 1 4 ‖x‖ 2 dΣ Hil´arioAlencar, Greg´ orio Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development - CNPq of Brazil. Date : June 16, 2021. 2020 Mathematics Subject Classification. Primary 53E10; 53C42; 53C21; Secondary 35C06; 35A15; 35A23; 35J15; 35J60. Key words and phrases. Self-shrinkers, stability, spectrum, index, drifted Laplacian, weighted mani- folds, f -minimal surfaces. 1