CyberSec Research

We prove a type of systolic inequality for embeddings of $T^2$ in $\mathbb{R}^3$. In particular, a highly twisted $T^2$ embedded in $\mathbb{R}^3$ must contain a non-contractible loop of small $\mathbb{R}^3$-diameter.

Riemannian Einstein solvmanifolds can be described in terms of nilsolitons, namely nilpotent Lie groups endowed with a left-invariant Ricci soliton metric. This characterization does not extend to indefinite metrics; nonetheless, nilsolitons can be defined and used to construct Einstein solvmanifolds of a higher dimension in any signature. An Einstein solvmanifold obtained by this construction turns out to satisfy the pseudo-Iwasawa condition, meaning that its Lie algebra splits as the orthogonal sum of a nilpotent ideal and an abelian subalgebra, the latter acting by symmetric derivations. We prove that the only pseudo-Iwasawa solvmanifolds that admit a Killing spinor, invariant or not, are the hyperbolic half-spaces.

In this note, we will give an positive answer to Pan-Rong's conjecture that for an open manifold with nonnegative Ricci curvature, if its universal cover has Euclidean volume growth, then its fundamental group is finitely generated. Moreover the fundamental group is virtually abelian. The same result has been given by H.Huang-X.Huang for dimension 4. In fact, we will show the fundamental group finitely generated in a more general condition.

Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. A rigidity statement is also provided for $\mathsf{RCD}^{\star}(K,N)$ spaces. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity.

We explore the information geometry of L\'evy processes. As a starting point, we derive the $\alpha$-divergence between two L\'evy processes. Subsequently, the Fisher information matrix and the $\alpha$-connection associated with the geometry of L\'evy processes are computed from the $\alpha$-divergence. In addition, we discuss statistical applications of this information geometry. As illustrative examples, we investigate the differential-geometric structures of various L\'evy processes relevant to financial modeling, including tempered stable processes, the CGMY model, and variance gamma processes.

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous K\"ahler manifolds. As a first immediate consequence we show the triviality of a K\"ahler-Ricci soliton submanifod of $C \times \Omega$, where $C$ is a flag manifold and $\Omega$ is a homogeneous bounded domain. Secondly, we show that no \emph{weak-relative} relationship can occur among the fundamental classes of homogeneous K\"ahler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two K\"ahler manifolds are said to be \emph{weak relatives} if they share, up to local isometry, a common K\"ahler submanifold of complex dimension at least two. Our main result precisely shows that if $E$ is (possibly indefinite) flat, $C$ is a flag manifold, and $\Omega$ is a homogeneous bounded domain, then: $E$ is not weak relative to $C\times\Omega$; $C$ is not weak relative to $E\times\Omega$; $\Omega$ is not weak relative to $E\times C$. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from \emph{relatives} to the more flexible notion of \emph{weak relatives} and dispense with the earlier ''special'' restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].

Certain data are naturally modeled by networks or weighted graphs, be they arterial networks or mobility networks. When there is no canonical labeling of the nodes across the dataset, we talk about unlabeled networks. In this paper, we focus on the question of dimensionality reduction for this type of data. More specifically, we address the issue of interpreting the feature subspace constructed by dimensionality reduction methods. Most existing methods for network-valued data are derived from principal component analysis (PCA) and therefore rely on subspaces generated by a set of vectors, which we identify as a major limitation in terms of interpretability. Instead, we propose to implement the method called barycentric subspace analysis (BSA), which relies on subspaces generated by a set of points. In order to provide a computationally feasible framework for BSA, we introduce a novel embedding for unlabeled networks where we replace their usual representation by equivalence classes of isomorphic networks with that by equivalence classes of cospectral networks. We then illustrate BSA on simulated and real-world datasets, and compare it to tangent PCA.

The Hessian structure, introduced by Shima(1976), is a geometric structure consisting of a pair $(\nabla,g)$ of an affine connection $\nabla$ and a Riemannian metric $g$ satisfying certain conditions. On the other hand, the Born structure, introduced by Freidel et al.(2014), is a strictly stronger geometric structure than an almost (para-)Hermitian structure. Marotta and Szabo(2019) proved that for a given manifold endowed with a pair $(\nabla, g)$, one can introduce an almost Born structure on the tangent bundle. In this article, we study the equivalence between the conditions that the pair $(\nabla, g)$ defines a Hessian structure, and that the induced almost Born structure is integrable.

We prove that every 3-sphere of positive Ricci curvature contains some embedded minimal surface of genus 2. We also establish a theorem for more general 3-manifolds that relates the existence of genus $g$ minimal surfaces to topological properties regarding the set of all embedded singular surfaces of genus $\leq g$.

This remark pertains to isometric embeddings endowed with certain geometric properties. We study two embeddings problems for the universal cover $M$ of an $n$-dimensional Riemannian torus $(\TT^n,g)$. The first concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$, and the second one seeks an isometric embdding of $M$ that is equivariant with respect to the deck transformation group of covering map. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $\TT^n$. However, we doubt whether these bounds are optimal.

We introduce the \emph{holomorphic $k$-systole} of a Hermitian metric on $\mathbb{C}P^n$, defined as the infimum of areas of homologically non-trivial holomorphic $k$-chains. Our main result establishes that, within the set of Gauduchon metrics, the Fubini-Study metric locally minimizes the volume-normalized holomorphic $(n-1)$-systole. As an application, we construct Gauduchon metrics on $\mathbb{C}P^2$ arbitrarily close to the Fubini-Study metric whose homological $2$-systole is realized by non-holomorphic chains.

We study the inextensibility problem of the spacetime at a future boundary point. We detect the inextensibility of the spacetime by the volume-distance-ratio asymptote of the timelike diamond approaching the future boundary point. The fundamental idea is to compare the asymptote with the one in Minkowski spacetime. By this idea, we establish the inextensibility criteria for both $C^{0,1}$ and $C^0$ regularities. As applications, we prove that i) $C^{0,1}$-inextensibility of the interior solution of the spherically self-similar naked singularity in the gravitational collapse of a massless scalar field constructed by Christodoulou. The key estimate is on the volume form of the interior solution of the naked singularity in a self-similar coordinate system. ii) $C^0$-inextensibility of the spatially flat FLRW spacetime with asymptotically linear scale factor $a(t) \sim t$. The key estimate is the volume comparison with the spatially hyperbolic FLRW spacetime with the scale factor $t$ which is the causal past of a point in the Minkowski spacetime.

The holonomy group of the adapted connection on a K-contact Riemannian manifold $(M, \theta, g)$ is considered. It is proved that if the orbit space $M/\xi$ of the Reeb field $\xi$ action admits a manifold structure, then the holonomy group of the adapted connection on $M$ is isomorphic to the holonomy group of the Levi-Civita connection on the Riemannian manifold $(M/\xi, h)$, where $h$ is the induced Riemannian metric on $M/\xi$. Thanks to this result, a simple proof of the de Rham theorem for the case of K-contact sub-Riemannian manifolds is obtained, stating that if the holonomy group of the adapted connection on $M$ is not irreducible, then the orbit space $M/\xi$ is locally a product of Riemannian manifolds.

Extending the recent work of Cannarsa, Cheng and Fathi, we investigate topological properties of the locus ${\cal NU}(M,g)$ of multiple maximizing geodesics on a globally hyperbolic spacetime $(M,g)$, i.e.\ the set of causally related pairs $(x,y)$ for which there exists more than one maximizing geodesic (up to reparametrization) from $x$ to $y$. We will prove that this set is locally contractible. We will also define the notion of a Lorentzian Aubry set ${\cal A}$ and prove that the inclusions ${\cal NU}(M,g)\hookrightarrow \operatorname{Cut}_M\hookrightarrow J^+\backslash {\cal A}$ are homotopy equivalences.

In this paper, we extend our investigation of the class of biconservative surfaces with non-constant mean curvature in 4-dimensional space forms $N^4(\epsilon)$. Specifically, we focus on biconservative surfaces with non-parallel normalized mean curvature vector fields (non-PNMC) that have flat normal bundles and are Weingarten. In our initial result we obtain the compatibility conditions for this class of biconservative surfaces in terms of an ODE system. Subsequently, by prescribing the flat connection in the normal bundle, we prove an existence result for the considered class of biconservative surfaces. Furthermore, we determine all non-PNMC biconservative Weingarten surfaces with flat normal bundles that either exhibit a particular form of the shape operator in the direction of the mean curvature vector field or have constant Gaussian curvature $K = \epsilon$. Finally, we prove that such surfaces cannot be biharmonic.

We construct universal geometric spaces over the real spectrum compactification $\Xi^{\mathrm{RSp}}$ of the character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $\Xi \times Y \rightarrow \Xi$ extends to a $\Gamma$-equivariant continuous surjection $(\Xi \times Y)^{\mathrm{RSp}} \rightarrow \Xi^{\mathrm{RSp}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{RSp}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y=\mathbb{P}^1$, the image is a real subtree.

A brief proof of Lie's classification of finite dimensional subalgebras of vector fields on the complex plane that have a proper Levi decomposition is given. The proof uses basic representation theory of sl(2, C). This, combined with \cite{ABF2} and \cite{ABF3} completes the classification of finite dimensional subalgebras of vector fields on the complex plane.

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to derive a rigidity result. Specifically, we obtain a complete characterization of all compact spacelike submanifolds with parallel mean curvature vector field that lie in the light cone of the Lorentz-Minkowski spacetime: they must be totally umbilical spheres contained in a spacelike hyperplane in Lorentz-Minkowski spacetime.