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M. Dajczer

Publications and source records attributed to M. Dajczer.

32 records · Page 2Linked to original sources
arXiv2017

Complete minimal submanifolds with nullity in the hyperbolic space

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ 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 \cite{dksv1} and \cite{dksv2}, 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 $\Hy^n$.

math.DG↗
arXiv2017

Complete minimal submanifolds with nullity in Euclidean spheres

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 \cite{harvey} and were initially studied by Bryant \cite{br}. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit \cite{DF2} 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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arXiv2017

Complete minimal submanifolds with nullity in Euclidean space

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let $M^m$ be a complete Riemannian manifold and let $f\colon M^m\to\R^n$ be a minimal isometric immersion with index of relative nullity at least $m-2$ at any point. We show that if the Omori-Yau maximum principle for the Laplacian holds on $M^m$, for instance, if the scalar curvature of $M^m$ does not decrease to $-\infty$ too fast or if the immersion $f$ is proper, then the submanifold must be a cylinder over a minimal surface.

math.DG↗
arXiv2015

A representation for pseudoholomorphic surfaces in spheres

We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman, we assign such a surface in $\Sf^{2n}$ to a given set of $n$ holomorphic functions defined on a simply-connected domain in $\C$.

math.DG↗
arXiv2015

Isometric deformations of isotropic surfaces

It was shown by Ramanathan \cite{R} that any compact oriented non-simply-connected minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. Here we show that this result extends to isotropic surfaces in spheres of arbitrary dimension. The case of non-compact isotropic surfaces in space forms is also addressed.

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arXiv2014

Euclidean hypersurfaces with a totally geodesic foliation of codimension one

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.

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arXiv2011

Isometric immersions of warped products

We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.

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arXiv2009

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature.

math.DG↗
arXiv2005

The vectorial Ribaucour transformation for submanifolds and applications

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of this result is the classical permutability of Ribaucour transformations. Our main application is an explicit local construction of all Euclidean submanifolds with flat normal bundle. Actually, this is a particular case of a more general result. Namely, we obtain a local explicit construction of all Euclidean submanifolds carrying a parallel flat normal subbundle, in particular of all those that carry a parallel normal vector field. Finally, we describe all submanifolds carrying a Dupin principal curvature normal vector field with integrable conullity, a concept that has proven to be crucial in the study of reducibility of Dupin submanifolds.

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arXiv2004

Isometric immersions in codimension two of warped products into space forms

We provide a local classification of isometric immersions $f\colon L^p\times_ρM^n\to\Q_c^{p+n+k}$ in codimensions $k=1, 2$ of warped products of Riemannian manifolds into space forms, under the assumptions that $n\geq k+1$ and that $N^{p+n}=L^p\times_ρM^n$ has no points with the same constant sectional curvature $c$ as the ambient space form.

math.DG↗