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The shift to smart grids has made electrical power systems more vulnerable to sophisticated cyber threats. To protect these systems, holistic security measures that encompass preventive, detective, and reactive components are required, even with encrypted data. However, traditional intrusion detection methods struggle with encrypted traffic, our research focuses on the low-level communication layers of encrypted power grid systems to identify irregular patterns using statistics and machine learning. Our results indicate that a harmonic security concept based on encrypted traffic and anomaly detection is promising for smart grid security; however, further research is necessary to improve detection accuracy.

In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in\mathbb{N}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k(\mathbb{S}^2\times \mathbb{S}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on $\mathbb{CP}^2\#\mathbb{CP}^2$. The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics., Comment: 160 pages, no figure, v2: added initial condition in some statements in the immortal case, proofs are unchanged

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth $n$-dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions., Comment: 34 pages, introduction slightly extended and references updated

We prove the existence of a unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two complex dimensions., Comment: 11 pages

Let $D$ be a toric K\"ahler-Einstein Fano manifold. We show that any toric shrinking gradient K\"ahler-Ricci soliton on certain toric blowups of $\mathbb{C}\times D$ satisfies a complex Monge-Amp\`ere equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method., Comment: 66 pages, various corrections, Proposition 7.15 revised

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by $c\cdot t^{-1}$ converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott., Comment: 41 pages, comments welcome!

We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient K\"ahler-Ricci soliton $(M,\,g,\,X)$ with soliton metric $g$ with bounded scalar curvature $\operatorname{R}_{g}$ whose soliton vector field $X$ has an integral curve along which $\operatorname{R}_{g}\not\to0$ is biholomorphic to either $\mathbb{C}\times\mathbb{P}^{1}$ or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the K\"ahler-Ricci flow on compact K\"ahler surfaces, leading to a classification of the bubbles of such singularities in this dimension., Comment: 32 pages; oversight in the proof of Theorem B rectified (uniform lower bound on volume of curves)

We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $\lambda$-functional adapted to the ALE situation and denoted $\lambda_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5., Comment: 56 pages, comments welcome!

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two such self-similar solutions to the Ricci flow is well-defined., Comment: 63 pages

We introduce a new functional inspired by Perelman's $\lambda$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $\lambda_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $\lambda_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $\lambda_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations., Comment: 63 pages, no figure

We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient K\"ahler-Ricci soliton in every K\"ahler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient K\"ahler-Ricci soliton on the cone., Comment: 92 pages

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)), x_0) \to (M,d_0, x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g(t)}(x_0,1) \Subset M$ for all $t\in (0,T)$ and $d_0$ is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution $\tilde g(t)_{t\in (0,T)}$ to the $\delta$-Ricci-DeTurck flow on an Euclidean ball ${\mathbb B}_{r}(p_0) \subset {\mathbb R}^n$, which can be extended to a smooth solution defined for $t \in [0,T)$. We further show, that this implies that the original solution $g$ can be extended to a smooth solution on $B_{d_0}(x_0,r/2)$ for $t\in [0,T)$, in view of the method of Hamilton., Comment: 37 pages, no figures. Journal version, to appear in JEMS. This version contains a small number of extra clarifications and explanations, partly resulting from comments of the referees

We first show that a K\"ahler cone appears as the tangent cone of a complete expanding gradient K\"ahler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient K\"ahler-Ricci solitons with quadratic curvature decay with derivatives. We then show that any two-dimensional complete shrinking gradient K\"ahler-Ricci soliton whose scalar curvature tends to zero at infinity is, up to pullback by an element of $GL(2,\,\mathbb{C})$, either the flat Gaussian shrinking soliton on $\mathbb{C}^{2}$ or the $U(2)$-invariant shrinking gradient K\"ahler-Ricci soliton of Feldman-Ilmanen-Knopf on the blowup of $\mathbb{C}^{2}$ at one point. Finally, we show that up to pullback by an element of $GL(n,\,\mathbb{C})$, the only complete shrinking gradient K\"ahler-Ricci soliton with bounded Ricci curvature on $\mathbb{C}^{n}$ is the flat Gaussian shrinking soliton and on the total space of $\mathcal{O}(-k)\to\mathbb{P}^{n-1}$ for $0

We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique., Comment: 31 pages. Final version, to appear in Math. Annalen

In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a $0$-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved.

In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities can be written as the divergences of genuine vector densities in the critical dimensions $D=2,4$, etc. As Lovelock gravity is a dimensional continuation of Euler densities, these results are of relevance for Gauss-Bonnet and, in general, Lovelock gravity. Indeed, these vectors which can be called generalized Katz vectors ensure, in particular, a well-posed Dirichlet variational principle., Comment: 15 pages

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschitz map that is homotopic to a constant., Comment: 37 pages ; Exposition of the paper re-organized ; The statement of the main Theorem has been changed to only allow target manifolds which are hypersurfaces of some Euclidean space. To appear in Annales de l'\'Ecole Normale Sup\'erieure

We propose to unify two a priori distinct aspects of black hole physics : their thermodynamics, and their effective dynamics when they are "skeletonized" as point particles (a useful procedure when tackling, for example, their motion in a coalescing binary system). For that purpose, the Einstein- Maxwell-dilaton (EMD) theory, which contains simple examples of asympotically flat, hairy black hole solutions, will serve as a laboratory. We will find that, when reducing a black hole to a point particle endowed with its specific, scalar-field-sensitive, effective mass, one in fact describes a black hole satisfying the first law of thermodynamics, such that its global charges, and hence its entropy, remain constant. This shows that the integration constant entering the scalar-field dependent mass is its entropy.

We propose a boundary term to the Einstein-Gauss-Bonnet action for gravity, which is constructed as the dimensional continuation of the Chern-Weil theorem, and such that the extremization of the full action yields the equations of motion when Dirichlet boundary conditions are imposed. When translated into tensorial language, this boundary term is the generalization to this theory of the Katz boundary term and vector for general relativity. The construction allows to deal with a general background and includes the Gibbons-Hawking-Myers boundary term as a special case, when the background is taken to be a product manifold. As a first application we show that this Einstein Gauss-Bonnet Katz action yields, without any extra ingredients or regularization, the expected mass of the Boulware-Deser black hole., Comment: 13 pages

We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable., Comment: 35 pages, final version, to appear in J. Geom. Anal

In this paper we address the two-body problem in massless Scalar-Tensor (ST) theories within an Effective-One-Body (EOB) framework. We focus on the first building block of the EOB approach, that is, mapping the conservative part of the two-body dynamics onto the geodesic motion of a test particle in an effective external metric. To this end, we first deduce the second post-Keplerian (2PK) Hamiltonian of the two-body problem from the known 2PK Lagrangian. We then build, by means of a canonical transformation a ST-deformation of the general relativistic EOB Hamiltonian which allows to incorporate the Scalar-Tensor (2PK) corrections to the currently best available General Relativity EOB results. This EOB-ST Hamiltonian defines a resummation of the dynamics that may provide information on the strong-field regime, in particular, the ISCO location and associated orbital frequency and can be compared to other, e.g. tidal, corrections.

We give necessary and sufficient conditions for a K\"ahler equivariant resolution of a K\"ahler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient K\"ahler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and for any negative line bundle $L$ over a compact K\"ahler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient K\"ahler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_{1}(K_{D}\otimes(L^{*})^{\otimes (n+1)})>0$. This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient K\"ahler-Ricci solitons on $\mathbb{C}^{n}$ with positive curvature operator on $(1,\,1)$-forms is path-connected., Comment: 44 pages. Corrected several typos. Refined Theorem A and Corollary B

In this paper we describe 4-dimensional gravity coupled to scalar and Maxwell fields by the Einstein-Katz action, that is, the covariant version of the "Gamma-Gamma $-$ Gamma-Gamma" part of the Hilbert action supplemented by the divergence of a generalized "Katz vector". We consider static solutions of Einstein's equations, parametrized by some integration constants, which describe an ensemble of asymptotically AdS black holes. Instead of the usual Dirichlet boundary conditions, which aim at singling out a specific solution within the ensemble, we impose that the variation of the action vanishes on shell for the broadest possible class of solutions. We will see that, when a long-range scalar "hair" is present, only sub-families of the solutions can obey that criterion. The Katz-Bicak-Lynden-Bell ("KBL") superpotential built on this (generalized) vector will then give straightforwardly the Noether charges associated with the spacetime symmetries (that is, in the static case, the mass). Computing the action on shell, we will see next that the solutions which obey the imposed variational principle, and with Noether charges given by the KBL superpotential, satisfy the Gibbs relation, the Katz vectors playing the role of "counterterms". Finally, we show on the specific example of dyonic black holes that the sub-class selected by our variational principle satisfies the first law of thermodynamics when their mass is defined by the KBL superpotential., Comment: Accepted for publication by JHEP

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity., Comment: 37 pages

We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymptotic cone., Comment: 29 pages

This article is an extended version of a paper presented in the WSOM'2012 conference [1]. We display a combination of factorial projections, SOM algorithm and graph techniques applied to a text mining problem. The corpus contains 8 medieval manuscripts which were used to teach arithmetic techniques to merchants. Among the techniques for Data Analysis, those used for Lexicometry (such as Factorial Analysis) highlight the discrepancies between manuscripts. The reason for this is that they focus on the deviation from the independence between words and manuscripts. Still, we also want to discover and characterize the common vocabulary among the whole corpus. Using the properties of stochastic Kohonen maps, which define neighborhood between inputs in a non-deterministic way, we highlight the words which seem to play a special role in the vocabulary. We call them fickle and use them to improve both Kohonen map robustness and significance of FCA visualization. Finally we use graph algorithmic to exploit this fickleness for classification of words.