Complete minimal submanifolds with nullity in Euclidean space

We investigate complete minimal submanifolds f : M 3 → H n in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in [6] and [7], respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of H n . of a geodesic sphere or an equidistant hypersurface or a horosphere if c > 0, c < 0 or c = 0, respectively. Let g : L 2 → Q n−ℓ c be an isometric immersion of a two-dimensional Riemannian manifold. The normal bundle of h = i • g : L 2 → H n splits orthogonally aswhere exp denotes the exponential map of H n . Then we denote by M m , m = 2 + ℓ, the open subset of N i Q n−ℓ c where G is an immersion endowed with the metric induced by the map G.Definition: The generalized cone in hyperbolic space over a surface g :1 Preliminaries Let f : M m → H n denote an isometric immersion of an m-dimensional Riemannian manifold M m into hyperbolic space. The relative nullity subspace of f at x ∈ M m is defined as D(x) = {X ∈ T x M : α(X, Y ) = 0 for all Y ∈ T x M} where α : T M × T M → N f M denotes the second fundamental form with values in the normal bundle.

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“…Proof: The proof is similar with those given in [6] and [7]. All we have to show is that V 2 does not contain regular points.…”

Section: From the Codazzi Equationsmentioning

confidence: 81%

“…We can assume that V 3 is empty since, otherwise, we already have by real analyticity that f is a totally geodesic submanifold. Proof: The proof goes as in [6] and [7]. Lemma 10.…”

Section: From the Codazzi Equationsmentioning

confidence: 99%

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

et al. 2018

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“…After excluding the latter case, we have from the real analyticity of f that V 3 is empty. We will proceed now following ideas developed in [10]. In fact, we only sketch the proof of the following fact, which is similar to the proof of Lemma 2 in [10].…”

Section: 3] the Set A Locally Decomposes Asmentioning

confidence: 99%

“…Let M m be a complete m-dimensional Riemannian manifold. In [10] we considered the case of minimal isometric immersions into Euclidean space f : M m → R n , m ≥ 3, satisfying that the index of relative nullity is at least m − 2 at any point. Under the mild assumption that the Omori-Yau maximum principle holds on M m , we concluded that any f must be "trivial", namely, just a cylinder over a complete minimal surface.…”

mentioning

confidence: 99%

Complete minimal submanifolds with nullity in Euclidean spheres

Dajczer

Kasioumis

Savas-Halilaj

et al. 2018

Comment. Math. Helv.

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In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m − 2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones.

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“…No Capítulo 4, combinando os resultados presentes em Asperti et al [1] e em Hasanis et al [17,18,19], enunciamos, assim como em [1], o Teorema 4.1 e o Teorema 4.2 que apresentam uma classificação de hipersuperfícies mínimas completas de Q 4 (c). Recentemente, Dajczer et al em [9,10,11] trabalharam com subvariedades mínimas e completas com nulidade em espaços forma.…”

Section: Introductionunclassified

A curvatura de Gauss-Kronecker de hipersuperfícies mínimas em espaços forma quadridimensionais

Santos¹

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In this work, we study the results obtained by Asperti et al. [1] and Hasanis et al. [17] involving the Gauss-Kronecker curvature of minimal hypersurfaces in four-dimensional space forms. We present concepts related to the study of Riemannian manifolds, as well as the orthonormal frame field technique used by both articles. Among the results of [1], a local version of the result obtained by Cheng [4] stands out for the Euclidean and hyperbolic cases. In the spherical case, we obtain an isometry between the image of a minimal immersion of a complete hypersurface with non-zero constant Gauss-Kronecker curvature and the Clifford torus. We also present two theorems referring to the classification of complete minimal hypersurfaces in four-dimensional space forms, in addition to developing the results found in [17].

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Complete minimal submanifolds with nullity in Euclidean space

Cited by 9 publications

References 29 publications

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

Complete minimal submanifolds with nullity in Euclidean spheres

A curvatura de Gauss-Kronecker de hipersuperfícies mínimas em espaços forma quadridimensionais

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