Minimal submanifolds with nullity in space forms (Doctoral thesis)

Κασιούμης, Θεόδωρος


In this thesis, we investigate complete minimal isometric immersions f: Mm-> Qc n into space forms with positive index of relative nullity. The index of relative nullity was introduced by Chern and Kuiper and turned out to be a fundamental concept in submanifold theory. At a point of Mm the index is just the dimension of the kernel of the second fundamental form of an isometric immersion f: Mm -> Qc n at that point. The kernels form an integrable distribution, the so called relative nullity distribution denoted by D, along any open subset where the index is constant and the images under f of the leaves of the foliation are affine subspaces in the ambient space. Our technique is to use the so called splitting tensor, which describes how the conullity distribution is twisting inside our manifold. We use tools from Geometric Analysis, like the Omori-Yau maximum principle and the gradient estimate of Yau in order to understand the structure of the splitting tensor. The main difficulty of the proof arises from the fact that we allow the index of relative nullity to vary. Hence, in order to extend the relative nullity distribution over the set A of totally geodesic points, we use regularity theorems for extending harmonic maps. At first, we consider complete minimal isometric immersions f: Mm -> Qc n into space forms Qc n, c=-1,0,1, with index of relative nullity at least m-2. Our technique for classifying the latter immersions consists of studing a tensor, the so called splitting tensor C, that describes how the conullity distribution is twisting inside the manifold Mm. We employ tools from geometric analysis, among them is the Omori-Yau maximum principle and the gradient estimate of Yau, in order to describe the structure of the splitting tensor as an endomorphism of the conullity distribution. The main difficulty arises from the fact that we allow the index of the relative nullity to vary. In order to extend the splitting tensor over the real analytic set A of totally geodesic points, it is essential to analyze the structure of the set A. This is accomplished by employing regularity extension theorems for harmonic maps. For minimal isometric immersions into Euclidean space Rn, we prove that the immersion f must be a cylinder over a minimal surface, under the mild assumption that the Omori-Yau maximum principle is satisfied for the Laplacian. The category of complete Riemannian manifolds for which the Omori-Yau maximum principle is valid is quite large. For instance, it contains the manifolds whose Ricci curvature is bounded from below. It also contains the class of properly immersed submanifolds in a space form whose norm of the mean curvature vector is bounded. The aforementioned result is truly global in nature, since there are plenty of non complete minimal submanifolds of dimension m having constant index of relative nullity m-2 that are not part of a cylinder on any open subset. They can all locally be parametrized in terms of a certain class of elliptic surfaces. Consequently, what remains as a challenging open problem is the existence of minimal complete and noncylindrical submanifolds with index of relative nullity at least m-2. It is worth noticing that many authors where interested into finding geometric conditions for an isometric immersion f: Mm -> Rn of a complete Riemannian mani-fold with positive index of relative nullity to be a cylinder. For complete minimal immersions f: Mm -> Sn in Euclidean spheres, we prove that any such submanifold Mm is either totally geodesic or has dimension three. In the latter case, there are plenty of examples, even compact ones. For any dimension and codimension there is an abundance of examples of noncomplete submanifolds fully described by Dajczer and Florit 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 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. In any of the two cases already studied, namely the Euclidean and spherical case, the proofs reduced to analyze the situation of the three dimensional submanifolds. In fact, for submanifolds in spheres only this case turned out to be possible. For minimal immersions f: Mm -> Hn in hyperbolic space of complete Riemannian manifolds Mm, the condition that the index of relative nullity is at least m-2 turns out to be quite less restrictive than in the previously studied cases. Nevertheless, we have reasons to believe that the manifold being three-dimensional is still quite special and this is why this case allows a characterization of a class of submanifolds that is contained in the following description. We prove that any three dimensional minimal submanifold f: M3 -> Hn having index of relative nullity at least one at any point, is either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of Hn, under the assumption that the scalar curvature is bounded from below. Furthermore, we parametrically describe all minimal immersions f: Mm -> Hn , whose index of relative nullity is m-2, as subbundles of the normal bundle of certain elliptic spacelike surfaces in the Lorentzian space or in the de Sitter space. From this parametrization it is straightforward than there exist a plethora of examples of non-complete minimal submanifolds with index of relative nullity m-2 . Additionally, using this parametrization,one can construct an abundance of complete minimal submanifolds of any dimension other than generalized cones. Finally we introduce a new class of minimal immersions F: Mn -> Hn+2, in the hyperbolic space that are (n-2)-ruled. This means that they carry an integrable tangent distribution of dimension n-2, whose leaves are mapped diffeomorphically by F onto open subsets of totally geodesic (n-2)- hyperbolic spaces of Hn+2. If the manifold is simply connected, we show that it allows a one-parameter family of equally ruled minimal isometric deformations that are genuine. The deformations are obtained while keeping fixed the normal bundle and the induced connection, but now the second fundamental form relates to the initial one in a much more complex form, in particular, no orthogonal tensor in involved. It is an interesting question if the above associated family of complete ruled minimal submanifolds exhausts all examples in the same class that admit genuine deformations. Of course, a much more challenging classification problem of complete submanifolds of rank four would be to drop one of the conditions, for instance being minimal or ruled.
Institution and School/Department of submitter: Πανεπιστήμιο Ιωαννίνων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών
Subject classification: Minimal submanifolds
Keywords: Minimal submanifolds,Index of relative nullity,Omori-Yau maximum principle,Real analytic set,Elliptic surfaces,Ruled submanifolds,Isometric minimal deformations,Ελαχιστικά υποπολυπτύγματα,Δείκτης μηδενοκατανομής,Αρχή μεγίστου,Αναλυτικά σύνολα,Ελλειπτικές επιφάνειες,Ευθειογενή υποπολυπτύγματα,Ελαχιστικές ισομετρικές παραμορφώσεις
URI: http://olympias.lib.uoi.gr/jspui/handle/123456789/29370
Appears in Collections:Διδακτορικές Διατριβές

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