128 research outputs found

### The structure of Gelfand-Levitan-Marchenko type equations for Delsarte transmutation operators of linear multi-dimensional differential operators and operator pencils. Part 1

An analog of Gelfand-Levitan-Marchenko integral equations for multi-
dimensional Delsarte transmutation operators is constructed by means of
studying their differential-geometric structure based on the classical Lagrange
identity for a formally conjugated pair of differential operators. An extension
of the method for the case of affine pencils of differential operators is
suggested.Comment: 12 page

### On the CR transversality of holomorphic maps into hyperquadrics

Let $M_\ell$ be a smooth Levi-nondegenerate hypersurface of signature $\ell$
in $\mathbf C^n$ with $n\ge 3$, and write $H_\ell^N$ for the standard
hyperquadric of the same signature in $\mathbf C^N$ with $N-n< \frac{n-1}{2}$.
Let $F$ be a holomorphic map sending $M_\ell$ into $H_\ell^N$. Assume $F$ does
not send a neighborhood of $M_\ell$ in $\mathbf C^n$ into $H_\ell^N$. We show
that $F$ is necessarily CR transversal to $M_\ell$ at any point. Equivalently,
we show that $F$ is a local CR embedding from $M_\ell$ into $H_\ell^N$.Comment: To appear in Abel Symposia, dedicated to Professor Yum-Tong Siu on
the occasion of his 70th birthda

### 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.Comment: First version (December 2008). Final version, including new title
(February 2009). To appear in Mathematische Annale

### Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic
hypersurfaces in complex spaces that are not transversally holomorphically
embeddable into hyperquadrics of any dimension. For this, we construct
invariants attached to a given hypersurface that serve as obstructions to
embeddability. We further study the embeddability problem for real-analytic
submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde

### A characterization of quadric constant mean curvature hypersurfaces of spheres

Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a
complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let
us denote by $\ell_v:M\to\mathbb{R}$ the function given by
$\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by
$f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove
that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some
real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ is
either a totally umbilical sphere or a Clifford hypersurface. As an
application, we will use this result to prove that the weak stability index of
any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$
which is neither totally umbilical nor a Clifford hypersurface and has constant
scalar curvature is greater than or equal to $2n+4$.Comment: Final version (February 2008). To appear in the Journal of Geometric
Analysi

### A global invariant for three dimensional CR-manifolds

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46621/1/222_2005_Article_BF01404456.pd

### Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations

The linearization of complex ordinary differential equations is studied by
extending Lie's criteria for linearizability to complex functions of complex
variables. It is shown that the linearization of complex ordinary differential
equations implies the linearizability of systems of partial differential
equations corresponding to those complex ordinary differential equations. The
invertible complex transformations can be used to obtain invertible real
transformations that map a system of nonlinear partial differential equations
into a system of linear partial differential equation. Explicit invariant
criteria are given that provide procedures for writing down the solutions of
the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single
research paper "Linearizability criteria for systems of two second-order
differential equations by complex methods" which has been published in
Nonlinear Dynamics. Due to citations of both parts I and II these are not
replaced with the above published articl

### Formal and finite order equivalences

We show that two families of germs of real-analytic subsets in $C^{n}$ are
formally equivalent if and only if they are equivalent of any finite order. We
further apply the same technique to obtain analogous statements for
equivalences of real-analytic self-maps and vector fields under conjugations.
On the other hand, we provide an example of two sets of germs of smooth curves
that are equivalent of any finite order but not formally equivalent

### A holonomy characterisation of Fefferman spaces

We prove that Fefferman spaces, associated to non--degenerate CR structures
of hypersurface type, are characterised, up to local conformal isometry, by the
existence of a parallel orthogonal complex structure on the standard tractor
bundle. This condition can be equivalently expressed in terms of conformal
holonomy. Extracting from this picture the essential consequences at the level
of tensor bundles yields an improved, conformally invariant analogue of
Sparling's characterisation of Fefferman spaces.Comment: AMSLaTeX, 15 page

### Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say
that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$}
($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal
$m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in
$\mathbb R^{m+k}$.
Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$
has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq
d-m$. The converse is not true in general.
We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in
$\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an
ordinary $\rho$-transversal. We use this result, together with the
multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann
category of the Grassmannian, and different versions of the colorful Helly
theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

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