CyberSec Research

We establish optimal stability estimates in terms of the Fraenkel asymmetry with universal dimensional constants for a Lorentzian isoperimetric inequality due to Bahn and Ehrlich and, as a consequence, for a special version of a Lorentzian isoperimetric inequality due to Cavalletti and Mondino. For the Bahn--Ehrlich inequality the Fraenkel asymmetry enters the stability result quadratically like in the Euclidean case while for the Cavalletti--Mondino inequality the Fraenkel asymmetry enters linearly. As it turns out, refining the latter inequality through an additional geometric term allows us to recover the more common quadratic stability behavior. Along the way, we provide simple self-contained proofs for the above isoperimetric-type inequalities.

In this paper we study the geometry and topology of compact Riemannian manifolds $(M,g)$ with boundary having the property that every geodesic that starts orthogonally to $\partial M$ also arrives orthogonally to the boundary.

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under a weighted null energy condition. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.

Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.

Pick $N$ random points $U_1,\cdots,U_{N}$ independently and uniformly in a triangle ABC with area 1, and take the convex hull of the set $\{A,B,U_1,\cdots,U_{N}\}$. The boundary of this convex hull is a convex chain $V_0=B,V_1,\cdots,$ $V_{\mathbf{n}(N)}$, $V_{\mathbf{n}(N)+1}=A$ with random size $\mathbf{n}(N)$. The first aim of this paper is to study the asymptotic behavior of this chain, conditional on $\mathbf{n}(N)=n$, when both $n$ and $m=N-n$ go to $+\infty$. We prove a phase transition: if $m=\lfloor n\lambda\rfloor$ where $\lambda>0$, this chain converges in probability for the Hausdorff topology to an (explicit) hyperbola ${\cal H}_\lambda$ as $n\to+\infty$, while, if $m=o(n)$, the limit shape is a parabola. We prove that this hyperbola is solution to an optimization problem: among all concave curves ${\cal C}$ in $ABC$ (incident with $A$ and $B$), ${\cal H}_\lambda$ is the unique curve maximizing the functional ${\cal C}\mapsto {\sf Area}({\cal C})^{\lambda} {\sf L}({\cal C})^3$ where ${\sf L}({\cal C})$ is the affine perimeter of ${\cal C}$. We also give the logarithm expansion of the probability ${\bf Q}^{\triangle \bullet\bullet}_{n,\lfloor n\lambda\rfloor}$, that $\mathbf{n}(N)=n$ when $N=n+\lfloor n\lambda\rfloor$. Take a compact convex set $\mathbf{K}$ with area 1 in the plane, and denote by ${\bf Q}^{\mathbf{K}}_{n,m}$ the probability of the event that the convex hull of $n+m$ iid uniform points in $\mathbf{K}$ is a polygon with $n$ vertices. We provide some results and conjectures regarding the asymptotic logarithm expansion of ${\bf Q}^{\mathbf{K}}_{n,m}$, as well as results and conjectures concerning limit shape theorems, conditional on this event. These results and conjectures generalize B\'ar\'any's results, who treated the case $\lambda=0$.

In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov--Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \begin{equation*} \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^{n-2}=\mathcal S, \end{equation*} where a point $z \in \mathcal S^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Minkowski dimension of $\mathcal S^k$ with respect to the spacetime distance is at most $k$. We show that the subset $\mathcal{S}^k_{\mathrm{qc}} \subset \mathcal{S}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum $\mathcal{S}^k$ is parabolic $k$-rectifiable for $k \in \{0, 1, 2\}$. Furthermore, we establish a sharp uniform $\mathscr{H}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of $\mathscr{H}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows.

We consider Lie groups equipped with left-invariant subbundles of their tangent bundles and norms on them. On these sub-Finsler structures, we study the normal curves in the sense of control theory. We revisit the Pontryagin Maximum Principle using tools from convex analysis, expressing the normal equation as a differential inclusion involving the subdifferential of the dual norm. In addition to several properties of normal curves, we discuss their existence, the possibility of branching, and local optimality. Finally, we focus on polyhedral norms and show that normal curves have controls that locally take values in a single face of a sphere with respect to the norm.

This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the model itself as a malleable geometric entity. Specifically, we optimize the metric tensor field on a manifold with a predefined topology, thereby dynamically shaping the geometric structure of the model space. To achieve this, we construct a variational framework whose loss function carefully balances data fidelity against the intrinsic geometric complexity of the manifold. The former ensures the model effectively explains observed data, while the latter acts as a regularizer, penalizing overly curved or irregular geometries to encourage simpler models and prevent overfitting. To address the computational challenges of this infinite-dimensional optimization problem, we introduce a practical method based on discrete differential geometry: the continuous manifold is discretized into a triangular mesh, and the metric tensor is parameterized by edge lengths, enabling efficient optimization using automatic differentiation tools. Theoretical analysis reveals a profound analogy between our framework and the Einstein-Hilbert action in general relativity, providing an elegant physical interpretation for the concept of "data-driven geometry". We further argue that even with fixed topology, metric optimization offers significantly greater expressive power than models with fixed geometry. This work lays a solid foundation for constructing fully dynamic "meta-learners" capable of autonomously evolving their geometry and topology, and it points to broad application prospects in areas such as scientific model discovery and robust representation learning.

In this paper we consider noncompact smooth metric measure spaces $(M, g,e^{-f}dvol_{g})$ of nonnegative Bakry-\'Emery Ricci curvature, i.e. $Ric + D^{2}f - \frac{1}{N}df \otimes df \geq 0$, for $0< N \leq \infty$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $\Omega\subset M$, with regular boundary $\partial \Omega$. Our inequalities are sharp for both the cases $N< \infty$ and $N= \infty$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\infty$ case holds true precisely when $M \setminus \Omega$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds.

In this paper, we classify Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form and three distinct principal curvatures. This completes the classification of such hypersurfaces initiated by Hu, Xie and Zhai in the article: Submanifolds with semi-parallel Mobius second fundamental form in the unit sphere. J.Geom. Anal. 33 (2023), 44pp.

We study geometric aspects of horizontal 2-plane distributions on the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in 3-dimensional projective space can be associated to such a geometric structure in 5-dimensions, and establish a dictionary between the projective differential geometry of the surface and the growth vector of the 2-plane distribution.

We prove well-posedness of the analytic Cauchy problem for gradient generalized Ricci solitons on an abelian bundle gerbe and solve the initial data equations on every compact Riemann surface. Along the way, we provide a novel characterization of the self-similar solutions of the generalized Ricci flow by means of families of automorphisms of the underlying abelian bundle gerbe covering families of diffeomorphisms isotopic to the identity.

In a recent preprint, Frank and the second author proved that if a metric on the sphere of dimension $d>4$ almost minimizes the total $\sigma_2$-curvature in the conformal class of the standard metric, then it is almost the standard metric (up to M\"obius transformations). This is achieved quantitatively in terms of a two-term distance to the set of minimizing conformal factors. We extend this result to the case $d=3$. While the standard metric still minimizes the total scalar curvature for $d=3$, it maximizes the total $\sigma_2$-curvature, which turns the related functional inequality into a reverse Sobolev-type inequality. As a corollary of our result, we obtain quantitative versions for a family of interpolation inequalities including the Andrews--De Lellis--Topping inequality on the $3$-sphere. The latter is itself a stability result for the well-known Schur lemma and is therefore called almost-Schur lemma. This makes our stability result an almost-almost-Schur lemma.

We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are representable by integration with respect to the normal cycle. This allows us to reduce the statement to a corresponding result for the representation of $\mathfrak{sl}(n)$ on the space of these differential forms.

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Amp\`ere-type equation and the associated Dirichlet problem.

This is an updated version of a paper which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as indicated in the text) that give references to descriptions of progress made in the field since the time of the original version of the paper. The main result of this paper is that every compact isoparametric hypersurface $M^n \subset S^{n+1} \subset {\bf R}^{n+2}$ is tight, i.e., every non-degenerate linear height function $\ell_p$, $p \in S^{n+1}$, has the minimum number of critical points on $M^n$ required by the Morse inequalities. Since $M^n$ lies in the sphere $S^{n+1}$, this implies that $M^n$ is also taut in $S^{n+1}$, i.e., every non-degenerate spherical distance function has the minimum number of critical points on $M^n$. A second result is that the focal submanifolds of isoparametric hypersurfaces in $S^{n+1}$ must also be taut. The proofs of these results are based on M\"{u}nzner's fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.

We study the holonomy that is associated to a sub-Riemannian structure defined on the kernel of a global contact form. This includes the holonomy of Schouten's horizontal connection as well as of the adapted connection, both canonical invariants of the structure. Under a condition on the torsion of the structure, we show that they are either equal or that the former is a codimension one normal subgroup of the latter. Furthermore, we establish a close relation to Riemannian holonomy, which yields a complete holonomy classification in the torsion-free case. For the main result we focus on the special case of pseudo-Hermitian structures and give a classification of holonomy algebras for both the Schouten and the adapted connection. Based on this, we derive a classification of symmetric sub-Riemannian structures and of of those holonomy groups that admit parallel spinors. Finally we exhibit a relation between locally symmetric sub-Riemannian contact structures and locally homogeneous Riemannian structures.

We study the variational behavior of the total inverse mean curvature of curves with prescribed boundary in the half-plane. We characterize the existence of critical points with prescribed area. We show that such critical points are strongly stable. As an application, we prove a local minimality property.