arXiv research

The paper classifies maximal translation surfaces in Lorentz-Minkowski space.

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

Unique maximal curve systems found for up to 5 punctures.

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

Interpolates curves using maximal and minimal surfaces in different spaces.

Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.

The paper explores fully affine maximal curves and their properties.

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Björling problem for the case of closed real analytic null curve. As an application, …

The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.

Classifies special quartic curves up to equivalence.

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

Maximizing energy on flexible curves yields regular or convex polygons.

Optimizes profit in targeted marketing across multiple markets with varying marketing expenditures.

New findings on hypersurfaces in Euclidean space that are both maximal and minimal.

This work studies learning curves for revenue maximization algorithms.

Generalizes symmetries of curved manifolds.

Extends Euler's problem to Lorentz-Minkowski plane.

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

Paper estimates optimal ROC curve arc length and AUC, improving classification performance.

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

We present a new mixture model-based discriminant analysis approach for functional data using a specific hidden process regression model. The approach allows for fitting flexible curve-models to each class of complex-shaped curves presenting regime changes. The model parameters are learned by maximizing the observed-da…

This study examines abnormal geodesics in 2D-Zermelo navigation problems, revealing their role in separating time minimal and maximal curves.

The curvature of web curves is studied in 3D manifolds.

This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associat…

This paper studies gradient flows in asymmetric metric spaces and proves existence results.

Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to $[2n/3]$ and the free rank…

We prove that every non-positively curved locally symmetric manifold M of finite volume contains a compact set K such that no periodic maximal flat can be homotoped out of K.

New proof of timelike minimal surfaces using split-harmonic maps.

Solves open problems on curved projective varieties.

We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.

Sparse curves on surfaces grow at a specific intermediate rate.

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the exis…

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…

We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…

Geodesics in curved spaces spread evenly over time.

Study of Moncrief lines' behavior in curved space-times.

New examples of translation surfaces on hyperelliptic curves with many automorphisms.

Study quotients of curve complex actions by mapping class group.

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compa…

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…

Support vector regression (SVR) is one of the most popular machine learning algorithms aiming to generate the optimal regression curve through maximizing the minimal margin of selected training samples, i.e., support vectors. Recent researchers reveal that maximizing the margin distribution of whole training dataset ra…

We extend the equivariant classification results of Escher and Searle for closed, simply connected, non-negatively curved Riemannian $n$-manifolds admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove…

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…