arXiv subjects

Niky Kamran

Publications and source records attributed to Niky Kamran.

At least 37 records · Page 2Linked to original sources

arXiv2019

The anisotropic Calder{ó}n problem on 3-dimensional conformally St{ä}ckel manifolds

Conformally St{ä}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$Δ$ G $ψ$ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n--1 commuting conformal symmetry operators for the Laplace-Beltrami operator $Δ$ G. In this paper, we solve the anisotropic Calder{ó}n problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally St{ä}ckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to dieomorphims that preserve the boundary.

math.AP↗

arXiv2019

On non-uniqueness for the anisotropic Calder{ó}n problem with partial data

We show that there is non-uniqueness for the Calder{ó}n problem with partial data for Riemannian metrics with H{ö}lder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only H{ö}lder continuous of order $$ρ$\<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M, g)$ which is not a warped product manifold, and we show that there exist in the conformal class of g an infinite number of Riemannian $\tilde{g} = c^4 g such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric g and do not satisfy the unique continuation principle.

math.AP↗

arXiv2018

Stability in the inverse Steklov problem on warped product Riemannian manifolds

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in R^d. We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

math.AP↗

arXiv2018

Spinors on Singular Spaces and the Topology of Causal Fermion Systems

Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples like the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kähler structures can be encoded in Riemannian fermion systems.

math-ph↗

arXiv2018

The anisotropic Calder{ó}n problem for singular metrics of warped product type: the borderline between uniqueness and invisibility

In this paper, we investigate the anisotropic Calder{ó}n problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics of (simple or double) warped product type, that is whose warping factors only depend on the horizontal direction of the cylinder. By singular, we mean that these factors are only assumed to be positive almost everywhere and to belong to some L p space with 0 < p $\le$ $\infty$. Using recent developments on the Weyl-Titchmarsh theory for singular Sturm-Liouville operators, we prove that the local Dirichlet-to-Neumann maps at each end are well defined and determine the metric uniquely if: 1. (Doubly warped product case) the coefficients of the metric are L $\infty$ and bounded from below by a positive constant. 2. (Warped product case) the coefficients of the metrics belong to a critical L p space where p < $\infty$ depends on the dimension of the compact fibers of the cylinder. Finally, we show (in the warped product case and for zero frequency) that these uniqueness results are sharp by giving simple counterexamples for a class of singular metrics whose coefficients do not belong to the critical L p space. All these counterexamples lead to a region of space that is invisible to boundary measurements.

math.AP↗

arXiv2018

A survey of non-uniqueness results for the anisotropic Calder{ó}n problem with disjoint data

After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $\ge$ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n $\ge$ 3, we first show that there exist in the conformal class of g an infinite number of Riemannian metrics gmetrics metrics g such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency coincide when the Dirichlet data $Γ$D and Neumann data $Γ$N are measured on disjoint sets and satisfy $Γ$D $\cup$ $Γ$N = $\partial$M. The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe type on (M, g) and arise from a natural but subtle gauge invariance of the Calder{ó}n when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n $\ge$ 3 to the anisotropic Calder{ó}n problem at fixed frequency with data on disjoint sets, which do not arise from this gauge invariance. They are given by cylindrical Riemannian manifolds with boundary having two ends, equipped with a suitably chosen warped product metric. This survey concludes with some remarks on the case of manifolds with corners.

math.AP↗

arXiv2017

On the hidden mechanism behind non-uniqueness for the anisotropic Calder{ó}n problem with data on disjoint sets

We show that there is generically non-uniqueness for the anisotropic Calderón problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary $(M,g)$ of dimension $n\geq 3$, there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}$ such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data $Γ_D$ and Neumann data $Γ_N$ are measured on disjoint sets and satisfy $\overline{Γ_D \cup Γ_N} \ne \partial M$. The conformal factors that lead to these non-uniqueness results for the anisotropic Calderón problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold $(M,g)$ and are associated to a natural but subtle gauge invariance of the anisotropic Calderón problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension $n\geq 3$ to the anisotropic Calderón problem at fixed frequency with data on disjoint sets and \emph{modulo this gauge invariance}. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.

math.AP↗

arXiv2017

Local isometric immersions of pseudo-spherical surfaces and k-th order evolution equations

We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature-1. We investigate the following problem: given such a metric, is there a local isometric immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for second order evolution equation to k-th order equations, we prove that there is only one type of equations that admit such an isometric immersion. We prove that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions u are universal, i.e., they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

math.DG↗

arXiv2017

Lorentzian Einstein metrics with prescribed conformal infinity

We prove a local well-posedness theorem for the (n+1)-dimensional Einstein equations in Lorentzian signature, with initial data $(\tilde g, K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\hat g$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an n-dimensional asymptotically hyperbolic Riemannian manifold $(M,\tilde g)$ such that the conformally rescaled metric $x^2 \tilde g$ (with $x$ a boundary defining function) extends to the closure $\bar M$ of $M$ as a metric of class $C^{n-1}$ which is also polyhomogeneous of class $C^{p}$ on $\bar M$. Likewise we assume that the conformally rescaled symmetric (0,2)-tensor $x^{2}K$ extends to the closure as a tensor field of class $C^{n-1}$ which is polyhomogeneous of class $C^{p-1}$. We assume that the initial data $(\tilde g, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M\times (-T_0,T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension n, such that if $p \geq 2q+r_n$, with $q$ a positive integer, then there is $T>0$, depending only on the norms of the initial and boundary data, such that the Einstein equations have a unique (up to a diffeomorphism) solution $g$ on $(-T,T)\times M$ with the above initial and boundary data, which is such that $x^2g$ is of class $C^{n-1}$ and polyhomogeneous of class $C^q$. Furthermore, if $x^2\tilde g$ and $x^2K$ are polyhomogeneous of class $C^\infty$ and $\hat g$ is in $C^\infty$, then $x^2g$ is polyhomogeneous of class $C^\infty$.

math.AP↗

arXiv2015

Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets

In this paper, we give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds.

math-ph↗

arXiv2015

Determining an asymptotically AdS spacetime from data on its conformal boundary

An outstanding question lying at the core of the AdS/CFT correspondence in string theory is the holographic prescription problem for Einstein metrics, which asserts that one can slightly perturb the conformal geometry at infinity of the anti-de Sitter space and still obtain an asymptotically anti-de Sitter spacetime that satisfies the Einstein equations with a negative cosmological constant. The purpose of this paper is to address this question by providing a precise quantitative statement of the real-time holographic principle for Einstein spacetimes, to outline its proof and to discuss its physical implications.

gr-qc↗

arXiv2015

Second-order equations and local isometric immersions of pseudo-spherical surfaces

We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u\_t=F(u,u\_x,u\_{xx})$ and $u\_{xt}=F(u, u\_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat \cite{RabeloTenenblat90}. We answer the following question: Given a pseudo-spherical surface determined by a solution $u$ of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in $\mathbb{R}^3$ depend on a jet of finite order of $u$? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of $x$ and $t$, independent of $u$.

math.DG↗

arXiv2015

Gaussian measures on the of space of Riemannian metrics

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance between Riemannian metrics and give applications to the diameter, eigenvalue and volume entropy functionals.

math.DG↗

arXiv2015

Third order differential equations and local isometric immersions of pseudospherical surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in ${\bf E}^{3}$ of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the pde. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In the present paper, we consider third-order equations of the form $u_{t}-u_{xxt}=λu u_{xxx} + G(u,u_x,u_{xx}),\, λ\in \mathbb{R}, $ which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in ${\bf E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, then these coefficients are universal in the sense of being independent on the choice of solution $u$.

math.DG↗

arXiv2015

Local Isometric immersions of pseudo-spherical surfaces and evolution equations

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in ${\bf E}^{3}$ from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in ${\bf E}^{3}$. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper [11], we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for $u(x,t)$ of order $k\geq 3$ describing pseudo-spherical surfaces. We show that whenever an isometric immersion in ${\bf E}^3$ exists, depending on a jet of finite order of $u$, then the coefficients of the second fundamental forms are functions of the independent variables $x$ and $t$ only.

math.DG↗

arXiv2014

Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of $2 \times 2$ matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we show that the knowledge of the reflection operators at a fixed non zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration.

math-ph↗

arXiv2013

A singular initial-boundary value problem for nonlinear wave equations and holography in asymptotically anti-de Sitter spaces

We analyze the initial value problem for semilinear wave equations on asymptotically anti-de Sitter spaces using energy methods adapted to the geometry of the problem at infinity. The key feature is that the coefficients become strongly singular at infinity, which leads to considering nontrivial data on the conformal boundary of the manifold. This question arises in Physics as the holographic prescription problem in string theory.

math.AP↗

arXiv2013

An Exact Expression for Photon Polarization in Kerr Geometry

We analyze the transformation of the polarization of a photon propagating along an arbitrary null geodesic in Kerr geometry. The motivation comes from the problem of an observer trying to communicate quantum information to another observer in Kerr spacetime by transmitting polarized photons. It is essential that the observers understand the relationship between their frames of reference and also know how the photon's polarization transforms as it travels through Kerr spacetime. Existing methods to calculate the rotation of the photon polarization (Faraday rotation) depend on choices of coordinate systems, are algebraically complex and yield results only in the weak-field limit. We give a closed-form expression for a parallel propagated frame along an arbitrary null geodesic using Killing-Yano theory, and thereby solve the problem of parallel transport of the polarization vector in an intrinsic, geometrically-motivated fashion. The symmetries of Kerr geometry are utilized to obtain a remarkably compact expression for the geometrically induced phase of the photon's polarization. We show that this phase vanishes on the equatorial plane and the axis of symmetry.

math-ph↗