arXiv research

Characterizes minor-minimal separating projective planar graphs and their generalizations.

The study restricts stable minimal immersions in product spaces to specific configurations.

Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.

The paper characterizes coverings over the projective plane with minimal defect.

The study constructs minimal submanifolds in complex and quaternionic projective spaces.

Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.

Study energy-minimizing maps in projective spaces, proving sharp bounds.

The paper shows how certain complex projective varieties can be broken down into simpler types.

Survey on minimal rational curves and their geometric structures.

Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.

New inequality linking geodesic length and volume in complex projective plane.

Study minimal Lagrangian surfaces in complex projective plane, focusing on contractible cases.

New IPL graphs identified and conditions for their projective embeddings established.

The paper classifies minimal projective varieties satisfying a specific equality.

In this paper we completely classify the homogeneous two-spheres, especially, the minimal homogeneous ones in the quaternionic projective space $\textbf{HP}^n$. According to our classification, more minimal constant curved two-spheres in $\textbf{HP}^n$ are obtained than Ohnita conjectured in the paper "Homogeneous har…

Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

Minimal equators and homological systoles found in Berger projective spaces.

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …

The study classifies stable submanifolds in product spaces of projective spaces.

In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using Kähler-Ricci flow.

Study on totally real flat minimal surfaces in quaternionic projective space.

The $\mathcal D$-groupoid of symmetries is minimal under specific conditions.

In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.

New Zoll families of minimal spheres found in spheres and projective spaces.

The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield …

We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other …

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Peterse…

A projection maps geodesic currents to Teichmüller space.

The paper characterizes minimal surfaces and Lagrangian surfaces in complex projective space.

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal t…

The study finds lower bounds for the warping degree of a knot projection.

The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…

In this paper, we study totally real minimal surfaces in the quaternionic projective space $\mathbb{H}P^n$. We prove that the linearly full totally real flat minimal surfaces of isotropy order $n$ in $\mathbb{H}P^n$ are two surfaces in $\mathbb{C}P^n$, one of which is the Clifford solution, up to symplectic congruence.

The paper simplifies knot and link diagrams with triple-crossings.

Optimizes energy of mappings from complex projective spaces.

Study 1-flat G-structures on uniruled projective manifolds.

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

Simply-connected surfaces of general type for n≥5.

We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.

This paper deals with unsupervised clustering with feature selection. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the sparsity of the matrix of weights, we propose an alternating minimization of the Fr…

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those ma…

Optimizes reinsurance and investment strategies to minimize ruin probability.

Optimal inequalities for metric surfaces derived from filling minimality.

Veronese minimizes normal curvatures to sphere.

We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of…