Characteristic Hypersurfaces Research Articles

Fano hypersurfaces in positive characteristic

Fano hypersurfaces in positive characteristic

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

FFRT properties of hypersurfaces and their F-signatures

This paper studies properties of certain hypersurfaces in prime characteristic: we give sufficient and necessary conditions for some classes of such hypersurfaces to have Finite [Formula: see text]-representation Type (FFRT) and we compute the [Formula: see text]-signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations.

Unique continuation through transversal characteristic hypersurfaces

We prove a unique continuation result for an ill-posed characteristic problem. A model problem of this type occurs in the A. D. Ionescu and S. Klainerman article (Theorem 1.1 in [11]) and we extend their model-result using only geometric assumptions. The main tools are Carleman estimates and Hörmander’s pseudo-convexity conditions.

Computing zeta functions of generic projective hypersurfaces in larger characteristic

We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the space complexity to $\tilde{O}(\log(p))$ while remaining polynomial in the other input parameters.

Method for generating of Archimedian triangular connective operators

Archimedean triangular operators form the basis of the tools for development, analysis and designing of fuzzy systems. They are generated by simple, monotone, single-valued and continuous functions. The basic sets of logical connectives of fuzzy systems are analyzed. It is shown that they are divided into conditional and algebraic ones. The well-known methods for generating of Archimedean triangular operators are described. Limitation of the functional characteristics of such generated operators is shown. To expand them, a method for generating has been created, which makes it possible to build new parameterized operators. Examples of constructing both new operators and those that generalize the already known ones are given. The tendency of the change of the characteristic hyper surface, which is build by new operator, is revealed.

On Global Dynamics of Three Dimensional Magnetohydrodynamics: Nonlinear Stability of Alfvén Waves

Magnetohydrodynamics (MHD) studies the dynamics of magnetic fields in electrically conducting fluids. In addition to the sound wave and electromagnetic wave behaviors, magneto-fluids also exhibit an interesting phenomenon: They can produce the Alfven waves, which were first described in a physics paper by Hannes Alfven in 1942. Subsequently, Alfven was awarded the Nobel prize for his fundamental work on MHD with fruitful applications in plasma physics, in particular the discovery of Alfven waves. This work studies (and constructs) global solutions for the three dimensional incompressible MHD systems (with or without viscosity) in strong magnetic backgrounds. We present a complete and self-contained mathematical proof of the global nonlinear stability of Alfven waves. Specifically, our results are as follows: We obtain asymptotics for global solutions of the ideal system (i.e.,viscosity \(\mu =0\)) along characteristics; in particular, we have a scattering theory for the system. We construct the global solutions (for small viscosity \(\mu \)) and we show that as \(\mu \rightarrow 0\), the viscous solutions converge in the classical sense to the zero-viscosity solution. Furthermore, we have estimates on the rate of the convergence in terms of \(\mu \). We explain a linear-driving decay mechanism for viscous Alfven waves with arbitrarily small diffusion. More precisely, for a given solution, we exhibit a time \(T_{n_0}\) (depending on the profile of the datum rather than its energy norm) so that at time \(T_{n_0}\) the \(H^2\)-norm of the solution is small compared to \(\mu \) (therefore the standard perturbation approach can be applied to obtain the convergence to the steady state afterwards). The results and proofs have the following main features and innovations: We do not assume any symmetry condition on initial data. The size of initial data (and the a priori estimates) does not depend on viscosity \(\mu \). The entire proof is built upon the basic energy identity. The Alfven waves do not decay in time: the stable mechanism is the separation (geometrically in space) of left- and right-traveling Alfven waves. The analysis of the nonlinear terms are analogous to the null conditions for non-linear wave equations. We use the (hyperbolic) energy method. In particular, in addition to the use of usual energies, the proof relies heavily on the energy flux through characteristic hypersurfaces. The viscous terms are the most difficult terms since they are not compatible with the hyperbolic approach. We obtain a new class of space-time weighted energy estimates for (weighted) viscous terms. The design of weights is one of the main innovations and it unifies the hyperbolic energy method and the parabolic estimates. The approach is ‘quasi-linear’ in nature rather than a linear perturbation approach: the choices of the coordinate systems, characteristic hypersurfaces, weights and multiplier vector fields depend on the solution itself. Our approach is inspired by Christodoulou–Klainerman’s proof of the nonlinear stability of Minkowski space-time in general relativity.

The Cauchy problem for the Pavlov equation with large data

We prove a local solvability of the Cauchy problem for the Pavlov equation with large initial data by the inverse scattering method. The Pavlov equation arises in studies Einstein–Weyl geometries and dispersionless integrable models. Our theory yields a local solvability of Cauchy problems for a quasi-linear wave equation with a characteristic initial hypersurface.

The Optical Properties of Gravity

The resemblance between the equation for a characteristic hypersurface through which wavefronts of light rays pass and optical metrics of general relativity has long been known. Discontinuities in the hypersurface are due to refraction involving Snell’s law, as opposed to discontinuities in time that would involve the Doppler effect. The presence of a static gravitational potential in the metric coefficients is accounted by an index of refraction that is entirely dependent on the space coordinates. The two-time Einstein metric must be reinterpreted as a two-space scale metric because of the two different speeds of light. It is shown that the Schwarzschild metric is incompatible with the laws of classical physics. Gravitational waves are identified with the transverse-trans-verse plane wave solutions to Einstein’s equations in vacuum, which propagate at the speed of light. Yet, when energy loss is evaluated, his equations acquire, surprisingly, a source term. Poynting’s vector, which is not a true vector, is defined in terms of the pseudo-gravitational tensor, and hence energy is neither localizable nor conserved. The solutions to the equations of motion are geodesics and, by definition, do not radiate. A finite speed of propagation implies that gravitational waves should aberrate, like their electromagnetic wave counterparts, and if they do not aberrate they cannot radiate.

Superluminal graviton propagation

We use the method of characteristics to study superluminal graviton (thereof called swift graviton) propagation in theories of higher curvature gravity of the form $($Riemann$)^2$, $($Riemann$)^3$, $\nabla^2 ($Riemann$)^2$ and $($Riemann$)^4$. We consider a pp-wave background. When probed by gravitons with an appropriate polarisation, several of the gravitational theories under consideration exhibit characteristic hypersurfaces outside the flat spacetime light-cone.

On the formation of shocks for quasilinear wave equations

The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation where $$G^{\prime \prime }(0)$$ is a non-zero constant. We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation. Since $$(\star )$$ satisfies the classical null condition, it admits global smooth solutions for small data. Therefore, we will work with large data (in energy norm). Moreover, no symmetry condition is imposed on the initial datum. We emphasize the geometric perspectives of shock formation in the proof. More specifically, the key idea is to study the interplay between the following two objects: (1) the energy estimates of the linearized equations of $$(\star )$$ ; (2) the differential geometry of the Lorentzian metric $$g=-\frac{1}{(1+3G^{\prime \prime }(0) (\partial _t\phi )^2)} d t^2+dx_1^2+dx_2^2+dx_3^2$$ . Indeed, the study of the characteristic hypersurfaces (implies shock formation) is the study of the null hypersurfaces of g. The techniques in the proof are inspired by the work (Christodoulou in The Formation of Shocks in 3-Dimensional Fluids. Monographs in Mathematics, European Mathematical Society, 2007) in which the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established. We also use the short pulse method which is introduced in the study of formation of black holes in general relativity in Christodoulou (The Formation of Black Holes in General Relativity. Monographs in Mathematics, European Mathematical Society, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012).

Degeneration of Fermat hypersurfaces in positive characteristic

We work over an algebraically closed field $k$ of positive characteristic $p$. Let $q$ be a power of $p$. Let $A$ be an $(n+1)\times(n+1)$ matrix with coefficients $a_{ij}$ in $k$, and let $X_A$ be a hypersurface of degree $q + 1$ in the projective space $\mathbf{P}^n$ defined by $\sum a_{ij}x_i x^q_j=0$. It is well-known that if the rank of $A$ is $n + 1$, the hypersurface $X_A$ is projectively isomorphic to the Fermat hypersuface of degree $q + 1$. We investigate the hypersurfaces $X_A$ when the rank of $A$ is $n$, and determine their projective isomorphism classes.

The cosmological constant and the energy of gravitational radiation

We propose a definition of mass for characteristic hypersurfaces in asymptotically vacuum space-times with non-vanishing cosmological constant $\Lambda \in {\mathbb R}^*$, generalising the definition of Trautman and Bondi for $\Lambda=0$. We show that our definition reduces to some standard definitions in several situations. We establish a balance formula linking the characteristic mass and a suitably defined renormalised volume of the null hypersurface, generalising the positivity identity of one of us (PTC) and Paetz proved when $\Lambda=0$.

Initial Value Problems for Wave Equations on Manifolds

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hörmander.

Causal structure in the scalar–tensor theory with field derivative coupling to the Einstein tensor

We investigate the causal structure in the scalar–tensor theory with the field derivative coupling to the Einstein tensor, which is a class of the Horndeski theory in the four-dimensional spacetime. We show that in general the characteristic hypersurface is non-null, which admits the superluminal propagations. We also derive the conditions that the characteristic hypersurface becomes null and show that a Killing horizon can be the causal edge for all the propagating degrees of freedom, if the additional conditions for the scalar field are satisfied. Finally, we find the position of the characteristic hypersurface in the dynamical spacetime with the maximally symmetric space, and that the fastest propagation can be superluminal, especially if the coupling constant becomes positive. We also argue that the superluminality itself may not lead to the acausality of the theory.

Shock formation in Lovelock theories

We argue that Lovelock theories of gravity suffer from shock formation, unlike General Relativity. We consider the propagation of (i) a discontinuity in curvature, and (ii) weak, high frequency, gravitational waves. Such disturbances propagate along characteristic hypersurfaces of a "background" spacetime and their amplitude is governed by a transport equation. In GR the transport equation is linear. In Lovelock theories, it is nonlinear and its solutions can blow up, corresponding to the formation of a shock. We show that this effect is absent in some simple cases e.g. a flat background spacetime, and demonstrate its presence for a plane wave background. We comment on weak cosmic censorship, the evolution of shocks, and the nonlinear stability of Minkowski spacetime, in Lovelock theories.

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.

Causality and hyperbolicity of Lovelock theories

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalize a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.

Mimicking acceleration in the constant-bang-time Lemaître-Tolman model: Shell crossings, density distributions, and light cones

The Lema\^{\i}tre--Tolman model with $\mathrm{\ensuremath{\Lambda}}=0$ and constant bang time that imitates the luminosity distance--redshift relation of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model using the energy function $E$ alone contains shell crossings. In this paper, the location in spacetime and the consequences of existence of the shell-crossing set (SCS) are investigated. The SCS would come into view of the central observer only at $t\ensuremath{\approx}1064T$ to the future from now, where $T$ is the present age of the Universe, but would not leave any recognizable trace in her observations. Light rays emitted near to the SCS are blueshifted at the initial points, but the blueshift is finite, and is overcompensated by later-induced redshifts if the observer is sufficiently far. The local blueshifts cause that $z$ along a light ray is not a monotonic function of the comoving radial coordinate $r$. As a consequence, the angular diameter distance ${D}_{A}$ and the luminosity distance ${D}_{L}$ from the central observer fail to be functions of $z$; the relations ${D}_{A}(z)$ and ${D}_{L}(z)$ are multiple-valued in a vicinity of the SCS. The following quantities are calculated and displayed: (1) The distribution of mass density on a few characteristic hypersurfaces of constant time; some of them intersect the SCS. (2) The distribution of density along the past light cone of the present central observer. (3) A few light cones intersecting the SCS at characteristic instants. (4) The redshift profiles along several light cones. (5) The extremum-redshift hypersurface. (6) The ${D}_{A}(z)$ and ${D}_{L}(z)$ relations. (7) The last scattering time and its comparison with the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ last scattering epoch.