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Abstract. Let ϕ : M m → N n be a minimal, proper immersion in an ambient space suitably close to a space form N n k 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 m k . 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. density and spectrum and Laplace-Beltrami and minimal submanifolds and monotonicity

In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.2010 Mathematics Subject Classification. Primary 53C42; Secondary 53A07, 53B20.

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the L r operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature H r +1 of the space forms N n+1 (c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of N n+1 (c) with H r +1 > 0 in terms of the r -th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero L reigenvalue of a closed hypersurface of N n+1 (c).

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