Density and spectrum of minimal submanifolds in space forms

Abstract

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

Mean curvature, volume and properness of isometric immersions

“…We can find in the very recent literature, (see Remark 4 in [24]), the proof that inequality (1.1) satisfied by the extrinsic balls D t of a minimal immersion ϕ : P m → N n in a Cartan-Hadamard manifold N n with sectional curvatures bounded from above K N ≤ b ≤ 0 implies the properness of P m . An older reference of the same result, but for minimal surfaces in R 3 with finite topology can be found in Theorem 3 in [9].…”

“…Concerning the techniques used, the proof in [24] is based in the intrinsic monotonicity property satisfied by the (intrinsic) volume growth…”

“…This approach inspires the interplay between the intrinsic and the extrinsic distances used in [24] to prove the properness of a minimal immersion in a Cartan-Hadamard manifold as well as our own way to prove Theorem 1.1, where we obtain a characterization of properness for non-minimal immersions in a broader family of ambient Riemannian manifolds, namely, the Riemannian manifolds with bounded geometry.…”

Asymptotically Extrinsic Tamed Submanifolds

“…However, this gap result does not hold for minimal submanifolds of the Hyperbolic space, as we can see in the following example, given in [22]. Example 1.4.…”

“…Example 1.4. In [22], the authors consider a minimal graph M n ⊆ H n+1 over a bounded and regular domain Ω ⊆ ∂ ∞ H n+1 , proving that M has finite total (extrinsic) curvature i.e. We can conclude from this fact that, in the case of minimal submanifolds of Hyperbolic space, to be extrinsically asymptotically flat (i.e., to have the curvature decay a(M ) = 0) it is not enough to characterize the hyperbolic subspaces, justifying the introduction of the invariant b(M ) and the extrinsic curvature decay criterion b(M ) < ∞.…”