CyberSec Research

In this paper, we establish a structure theorem for a compact K\"{a}hler manifold $X$ of rational dimension $\mathrm{rd}(X)\leq n-k$ under the mixed partially semi-positive curvature condition $\mathcal{S}_{a,b,k} \geq 0$, which is introduced as a unified framework for addressing two partially semi-positive curvature conditions -- namely, $k$-semi-positive Ricci curvature and semi-positive $k$-scalar curvature. As a main corollary, we show that a compact K\"{a}hler manifold $(X,g)$ with $k$-semi-positive Ricci curvature and $\mathrm{rd}(X)\leq n-k$ actually has semi-positive Ricci curvature and $\mathrm{rd}(X)\geq \nu(-K_X)$. Of independent interest, we also confirm the rational connectedness of compact K\"{a}hler manifolds with positive orthogonal Ricci curvature, among other results.

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,\omega)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,\omega)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),\omega_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, \omega_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

In 1985, Bryant stated that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if it satisfies a certain rationality condition. We extend this rationality criterion to arbitrary dimensional flat tori, providing a sufficient condition for minimal isometric immersions of flat $n$-tori. For the case $n=3$, we prove that if a flat $3$-torus admits a minimal isometric immersion into some sphere, then its algebraic irrationality degree must be no more than 4, and we construct explicit embedded minimal irrational flat $3$-tori realizing each possible degree. Furthermore, we establish the upper bound $n^2+n-1$ for the minimal target dimension of flat $n$-tori admitting minimal isometric immersions into spheres.

The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p}+V$ on high tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry is studied under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union of all local Landau levels of the operator at the points of $X$. Moreover, if the union of the local Landau levels over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete. The main result of the paper is the trace asymptotics formula associated with these eigenvalues. As a consequence, we get a Weyl type asymptotic formula for the eigenvalue counting function.

We demonstrate that the moduli space of Hermitian-Einstein connections $\text{M}^*_{HE}(M^{2n})$ of vector bundles over compact non-Gauduchon Hermitian manifolds $(M^{2n}, g, \omega)$ that exhibit a dilaton field $\Phi$ admit a strong K\"ahler with torsion structure provided a certain condition is imposed on their Lee form $\theta$ and the dilaton. We find that the geometries that satisfy this condition include those that solve the string field equations or equivalently the gradient flow soliton type of equations. In addition, we demonstrate that if the underlying manifold $(M^{2n}, g, \omega)$ admits a holomorphic and Killing vector field $X$ that leaves $\Phi$ also invariant, then the moduli spaces $\text{M}^*_{HE}(M^{2n})$ admits an induced holomorphic and Killing vector field $\alpha_X$. Furthermore, if $X$ is covariantly constant with respect to the compatible connection $\hat\nabla$ with torsion a 3-form on $(M^{2n}, g, \omega)$, then $\alpha_X$ is also covariantly constant with respect to the compatible connection $\hat D$ with torsion a 3-form on $\text{M}^*_{HE}(M^{2n})$ provided that $K^\flat\wedge X^\flat$ is a $(1,1)$-form with $K^\flat=\theta+2d\Phi$ and $\Phi$ is invariant under $X$ and $IX$, where $I$ is the complex structure of $M^{2n}$.

In this paper twistor methods are used to construct a family of multivalued harmonic functions on ${\bf R}^{3}$ which were obtained by Dashen Yan using different methods. The branching sets for the solutions are ellipses and the functions have quadratic growth at infinity.

By the results of Furuhata--Inoguchi--Kobayashi [Inf. Geom. (2021)] and Kobayashi--Ohno [Osaka Math. J. (2025)], the Amari--Chentsov $\alpha$-connections on the space $\mathcal{N}$ of all $n$-variate normal distributions are uniquely characterized by the invariance under the transitive action of the affine transformation group among all conjugate symmetric statistical connections with respect to the Fisher metric. In this paper, we investigate the Amari--Chentsov $\alpha$-connections on the submanifold $\mathcal{N}_0$ consisting of zero-mean $n$-variate normal distributions. It is known that $\mathcal{N}_0$ admits a natural transitive action of the general linear group $GL(n,\mathbb{R})$. We establish a one-to-one correspondence between the set of $GL(n,\mathbb{R})$-invariant conjugate symmetric statistical connections on $\mathcal{N}_0$ with respect to the Fisher metric and the space of homogeneous cubic real symmetric polynomials in $n$ variables. As a consequence, if $n \geq 2$, we show that the Amari--Chentsov $\alpha$-connections on $\mathcal{N}_0$ are not uniquely characterized by the invariance under the $GL(n,\mathbb{R})$-action among all conjugate symmetric statistical connections with respect to the Fisher metric. Furthermore, we show that any invariant statistical structure on a Riemannian symmetric space is necessarily conjugate symmetric.

Compact K\"ahler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for K\"ahler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a K\"ahler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.

We obtain a complete classification of ruled zero mean curvature surfaces in the three-dimensional light cone. En route, we examine geodesics and screw motions in the space form, allowing us to discover helicoids. We also consider their relationship to catenoids using Weierstrass representations of zero mean curvature surfaces in the three-dimensional light cone.

In this work, we discuss the stability of the pluriclosed flow and generalized Ricci flow. We proved that if the second variation of generalized Einstein--Hilbert functional is nonpositive and the infinitesimal deformations are integrable, the flow is dynamically stable. Moreover, we prove that the pluriclosed steady solitons are dynamically stable when the first Chern class vanishes.

We prove that every topological/smooth $\T=(\C^{*})^{n}$-equivariant vector bundle over a topological toric manifold of dimension $2n$ is a topological/smooth Klyachko vector bundle in the sense of arXiv:2504.02205.

The goal of this paper is to develop a theory of "sublinearly Morse boundary" and prove a corresponding sublinearly Morse lemma in higher rank symmetric space of non-compact type. This is motivated by the work of Kapovich-Leeb-Porti and the theory of sublinearly Morse quasi-geodesics developed in the context of CAT(0) geometry.

A Riemannian manifold is called a geodesic orbit manifolds, GO for short, if any geodesic is an orbit of a one-parameter group of isometries. By a result of C.Gordon, a non-flat GO nilmanifold is necessarily a two-step nilpotent Lie group with a left-invariant metric. We give a complete classification of non-singular GO nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and two one-parameter families of dimensions 14 and 15.

The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on $n$. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the $\text{SL}_n\mathbb{R}$ Hitchin component is a polyhedral complex of dimension at most $n-1$. For endpoints of cubic differential rays in the $\text{SL}_3\mathbb{R}$ Hitchin component, the dual space is the universal cover of $S$, equipped with an asymmetric Finsler metric which records growth rates of trace functions.

In this note, we show that (the germ of) each Euler-like vector field comes from a tubular neighborhood embedding given by the normal exponential map of some Riemannian metric.

For a minimal submanifold of the Euclidean space, we prove monotonicity formulas for its (weighted) volume within images of concentric balls under M\"obius transformations.

In this short note, we deal with Serrin-type problems in Riemannian manifolds. Firstly, we provide a Soap Bubble type theorem and rigidity results. In another direction, we obtain a rigidity result addressed to annular regions in Einstein manifolds endowed with a conformal vector field.

A new notion of quasilocal mass is defined for generic, compact, two dimensional, spacelike surfaces in four dimensional spacetimes with negative cosmological constant. The definition is spinorial and based on work for vanishing cosmological constant by Penrose and Dougan & Mason. Furthermore, this mass is non-negative, equal to the Misner-Sharp mass in spherical symmetry, equal to zero for every generic surface in AdS, has an appropriate form for gravity linearised about AdS and has an appropriate limit for large spheres in asymptotically AdS spacetimes.