Sharp Estimates for the First Stability Eigenvalue of Surfaces in the Presence of a Closed Conformal Vector Field
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Sharp Estimates for the First Stability Eigenvalue of Surfaces in the Presence of a Closed Conformal Vector Field ´ Miguel Angel Mero˜ no Abstract. The purpose of this article is to find out sharp estimates for the first eigenvalue of the stability operator for compact surfaces with constant mean curvature immersed into a Riemannian 3-manifold having a nowhere vanishing closed conformal vector field, and characterize the cases where the estimates are reached. Mathematics Subject Classification. 53C42, 53C40, 58C40. Keywords. Surface, Constant mean curvature, Stability eigenvalue, Closed conformal vector field.
1. Introduction Minimal surfaces into a Riemannian 3-manifold are the critical points of the area functional, whereas constant mean curvature (CMC) surfaces are the critical points of the same functional for variations that leave constant the enclosed volume. At a critical point, the stability of the corresponding variational problem is given by the second variation of the area functional, which gives to us the Jacobi or stability operator: J = Δ + |A|2 + Ric(N, N), where Δ stands for the Laplacian operator on the surface, A is the shape operator of the immersion, and Ric denotes the Ricci curvature of the ambient space (see [4]). The spectrum of J consists of an unbounded increasing sequence of eigenvalues with finite multiplicities, and the first one, denoted by λ1 , is special, since the surface is strongly stable if and only if λ1 ≥ 0. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Regi´ on de Murcia, Spain, by Fundaci´ on S´ eneca, Science and Technology Agency of the Regi´ on de Murcia. The author was partially supported by MICINN/FEDER project PGC2018-097046-B-I00 and Fundaci´ on S´ eneca project 19901/GERM/15, Spain.
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When we consider variations that leave constant the enclosed volume, the corresponding notion is known as weak stability. This unique feature makes λ1 play a key role in the study of the stability of CMC surfaces, and finding out estimates for it helps us in understanding such surfaces. In this regard, Simons [15] obtained an upper bound for λ1 on any minimal compact hypersurface in the standard sphere. In particular, for minimal surfaces in the 3-sphere he proved that λ1 = −2 if the surface is totally geodesic and λ1 ≤ −4 otherwise. Later on, Wu [16] characterized the equality by showing that it holds only for the minimal Clifford torus. More recently, Perdomo [14] gave a new proof of this spectral characterization by getting an interesting formula that relates the first eigenvalue λ1 , the genus of the surface, its area, and a simple invariant. Al´ıas, Barros and Brasil [1] extended Wu and Perdomo’s results to the case of CMC hypersurfaces in the standard sphere, characterizing some CMC Clifford tori. In [3], the authors studied the same problem for CMC compact surfaces immersed into homogeneous Riemannian 3-manifolds, finding out sharp upper bounds
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