Density and spectrum of minimal submanifolds in space forms

• Published in: Mathematische annalen Vol. 366; no. 3-4; pp. 1035 - 1066
• Main Authors: Lima, Barnabé Pessoa, Mari, Luciano, Montenegro, José Fabio Bezerra, Vieira, Franciane de Brito
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2016; Springer Nature B.V
• Subjects: Curvature; Density; Infinity; Manifolds (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; 35P15; 58C21; 58J50
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-016-1360-y

Let φ : M m → N n be a minimal, proper immersion in an ambient space suitably close to a space form N k n of curvature - k ≤ 0 . In this paper, we are interested in the relation between the density function Θ ( r ) of M and the spectrum of its Laplace–Beltrami operator. In particular, we prove that if Θ ( r ) has subexponential growth (when k < 0 ) or sub-polynomial growth ( k = 0 ) along a sequence, then the spectrum of M m is the same as that of the space form N k m . Notably, the result applies to Anderson’s (smooth) solutions of Plateau’s problem at infinity on the hyperbolic space, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures M to have finite density. In particular, we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density.