Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.
problem Understanding the Morse index of minimal hypersurfaces in real projective spaces.
method Analyzing unstable one-sided and two-sided minimal hypersurfaces in real projective spaces.
result The Morse index of minimal hypersurfaces is at least n+2, with specific examples provided.
The paper characterizes coverings over the projective plane with minimal defect.
problem Characterizing minimal defect branched coverings over the projective plane.
method Characterization through properties of decomposable and indecomposable coverings.
result Extended family of realizations and generalized results on primitive permutation groups.
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.
problem Uniqueness of generalized Gauss maps for minimal surfaces with shared hypersurfaces in projective varieties.
method Analysis of minimal surfaces in Rn+1 with inverse images of hypersurfaces in a projective subvariety. result Generalization and improvement of previous results on the uniqueness of generalized Gauss maps.
Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.
problem Characterizing minimal orbits of semi-simple Lie groups.
method Analyzing projective orbits induced by representations of semi-simple Lie groups and relating them to invariant subspaces of the underlying modules.
result Minimal orbits of semi-simple Lie groups are in bijection with minimal orbits of compact subgroups on invariant subspaces.
New inequality linking geodesic length and volume in complex projective plane.
problem Understanding geometric properties of complex projective plane.
method Combining recent results on area minimizers and geodesics with Kronheimer-Mrowka's proof.
result Proved a new inequality relating volume and length of geodesics.
Study minimal Lagrangian surfaces in complex projective plane, focusing on contractible cases.
problem Construct minimal Lagrangian surfaces in complex projective plane.
method Loop group method
result Presented new classes of minimal Lagrangian surfaces.
New IPL graphs identified and conditions for their projective embeddings established.
problem Characterizing and identifying intrinsically projectively linked graphs.
method Applying Δ-Y exchanges and analyzing projective planar graphs.
result No minor-minimal IPL graphs on 16 edges exist, and new ones are identified.
The paper classifies minimal projective varieties satisfying a specific equality.
problem Classifying minimal projective varieties with a specific equality.
method Established a structure theorem for minimal projective klt varieties satisfying Miyaoka's equality.
result Minimal projective klt varieties with Miyaoka's equality have semi-ample canonical divisors and specific Kodaira dimensions.
In this paper we completely classify the homogeneous two-spheres, especially, the minimal homogeneous ones in the quaternionic projective space HPn. According to our classification, more minimal constant curved two-spheres in HPn are obtained than Ohnita conjectured in the paper "Homogeneous har…
Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.
problem Constructing maps between Lagrangian submanifolds and quaternionic projective spaces.
method Explicit construction of maps from minimal δ(2)-ideal Lagrangian submanifolds of Cn to HPn−1. result One-to-one correspondences between minimal Lagrangian surfaces in CP2 and minimal totally complex surfaces in HP2. 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.
problem Tackles the minimality of real projective subspaces under Berger deformations.
method Uses the skew-adjoint endomorphism AV to classify minimal subspaces and compute homological systoles. result Equatorial hypersurfaces remain minimal, but not all real subspaces are minimal under Berger deformations.
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.
problem Classifying stable submanifolds in product spaces of projective spaces.
method Provided a classification theorem for compact stable minimal immersions in product spaces of projective spaces.
result Characterized complex minimal immersions in the product of two complex 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.
problem Characterizing the moduli space of totally real flat minimal immersions in HP^3.
method Analyzing the moduli space of linearly full totally real flat minimal immersions from C into HP^3.
result The moduli space has three components, each a 6-dimensional manifold.
The D-groupoid of symmetries is minimal under specific conditions.
problem Conditions for the minimality of the D-groupoid of symmetries of a projective structure. method Analyzing the D-groupoid and its sub-groupoids, and relating it to the non-integrability of certain equations. result The minimality of the D-groupoid is equivalent to the non-integrability of specific equations. 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.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
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 S3 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.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
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.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.
The paper characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
problem Characterizing minimal surfaces and Lagrangian surfaces in complex projective space.
method Using Ruh-Vilms type theorems.
result 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.
problem Determining the warping degree of a knot projection.
method Examining the maximal number of regions sharing no crossings for a fixed crossing in a knot projection.
result Lower bounds for the warping degree of a knot projection are provided.
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 HPn. We prove that the linearly full totally real flat minimal surfaces of isotropy order n in HPn are two surfaces in CPn, one of which is the Clifford solution, up to symplectic congruence.
The paper simplifies knot and link diagrams with triple-crossings.
problem Generating and classifying minimal triple-crossing knot and link diagrams.
method Systematic method to generate minimal triple-crossing projections, introducing new diagrammatic moves.
result Classification of knots and links with triple-crossing number up to five, derivation of minimal generating set of moves.
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Study 1-flat G-structures on uniruled projective manifolds.
problem Classify 1-flat irreducible G-structures on uniruled projective manifolds.
method Algebraic geometry and Cartan connections.
result Locally flat structures on VMRTs without 1-flatness assumption.
New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.
problem Finding non-bumpy metrics with multiplicity on spheres and projective spaces.
method New area-and-separation estimate for minimal hypersurfaces with Morse index two.
result First examples of non-bumpy metrics with multiplicity on (n+1)-spheres and projective spaces. Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected 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 Mn 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 RP2. We show that a minimal surface f:RP2→M3 which has the smallest area, among those ma…
Optimizes reinsurance and investment strategies to minimize ruin probability.
problem Optimizing reinsurance and investment strategies to minimize ruin probability.
method Stochastic projected gradient method based on Malliavin calculus.
result Effectiveness of the proposed method demonstrated through numerical experiments.
Optimal inequalities for metric surfaces derived from filling minimality.
problem Proving optimal systolic inequalities for metric surfaces.
method Analysis of asymptotic volume growth and minimality of normed planes and hemispheres.
result Optimal constants for tori and real projective planes match Finsler settings.
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees 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…