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

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 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

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

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

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

Topology, Entropy and Witten Index of Dilaton Black Holes

We have found that for extreme dilaton black holes an inner boundary must be introduced in addition to the outer boundary to give an integer value to the Euler number. The resulting manifolds have (if one identifies imaginary time) topology $S^1 \times R \times S^2$ and Euler number $\chi = 0$ in contrast to the non-extreme case with $\chi=2$. The entropy of extreme $U(1)$ dilaton black holes is already known to be zero. We include a review of some recent ideas due to Hawking on the Reissner-Nordstr\"om case. By regarding all extreme black holes as having an inner boundary, we conclude that the entropy of {\sl all} extreme black holes, including $[U(1)]^2$ black holes, vanishes. We discuss the relevance of this to the vanishing of quantum corrections and the idea that the functional integral for extreme holes gives a Witten Index. We have studied also the topology of ``moduli space'' of multi black holes. The quantum mechanics on black hole moduli spaces is expected to be supersymmetric despite the fact that they are not HyperK\"ahler since the corresponding geometry has torsion unlike the BPS monopole case. Finally, we describe the possibility of extreme black hole fission for states with an energy gap. The energy released, as a proportion of the initial rest mass, during the decay of an electro-magnetic black hole is 300 times greater than that released by the fission of an ${}^{235} U$ nucleus.Comment: 51 pages, 4 figures, LaTeX. Considerably extended version. New sections include discussion of the Witten index, topology of the moduli space, black hole sigma model, and black hole fission with huge energy releas

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