The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.
problem Characterizing semiaffine stable planes.
method Analyzing properties of lines and points in stable planes.
result Semiaffine stable planes are either affine, projective, or punctured projective planes.
Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP2. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
Study fake projective planes using recent results to show bicanonical map is always an embedding and construct an exceptional collection.
problem Analyzing Keum's fake projective planes and their geometric properties.
method Apply recent results from Galkin et al. [GKMS15] to study fake projective planes.
result The bicanonical map of Keum's fake projective planes is always an embedding.
Improved bounds on Laplace-Beltrami operator eigenvalues on real projective plane.
problem Improving upper bounds for Laplace-Beltrami operator eigenvalues.
method Analyzing eigenvalues with even indexes and providing bounds for Dirichlet, Neumann, and Steklov eigenvalues.
result Enhanced upper bounds for Laplace-Beltrami operator eigenvalues on the real projective plane.
A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.
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.
Study calculates rational homology groups of configuration spaces for a Moebius strip and a projective plane.
problem Calculating rational homology groups of configuration spaces for specific topological spaces.
method Explicit calculation of all rational homology groups.
result All rational homology groups of configuration spaces for the Moebius strip and projective plane are determined.
Researchers found parallel mean curvature tori in complex projective and hyperbolic planes.
problem Finding tori with parallel mean curvature vectors.
method Explicit determination of tori in complex projective and hyperbolic planes.
result Explicit determination of tori with parallel mean curvature vectors in both complex projective and hyperbolic planes.
Graphs mapped to projective plane classified up to deformation.
problem Classifying immersions of graphs to the projective plane.
method Classification via regular homotopy and construction of invariants.
result Complete invariant for equivalence classes of immersions.
Stable planes are locally isomorphic to classical projective planes.
problem Characterizing stable planes that are locally isomorphic to classical projective planes.
method Analyzing properties of stable planes and comparing them to classical projective planes over specific fields.
result Simply connected stable planes with connected lines are isomorphic to open subplanes of classical projective planes.
Smooth lens spaces embed in complex projective planes but not in finite copies.
problem Embedding lens spaces in complex projective planes.
method Analyzes smooth and locally flat embeddings of lens spaces in complex projective planes.
result No finite number of copies of complex projective plane can embed every lens space smoothly.
We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a (2,3)-elliptic surface that is the minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of Q-h…
The paper proves an inequality for the second non-zero eigenvalue of the Laplacian on the projective plane.
problem An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane.
method Analytical proof and geometric construction of limiting metrics.
result The second non-zero eigenvalue is not greater than 20π for metrics of unit area, with a specific limiting configuration.
Study of a 32D Rosenfeld projective plane, a symmetric space.
problem Understanding the properties of a specific symmetric space.
method Detailed study and analysis of the geometric structure.
result Comprehensive understanding of the 32D Rosenfeld projective plane.
Constructs projective plane over octonions, proving no higher real division algebras.
problem Existence of higher-dimensional real division algebras.
method Using Adams' solution of the Hopf invariant 1 problem, constructs projective plane over octonions.
result No higher-dimensional real division algebras exist.
Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
Computed p-widths for real projective plane.
problem Calculating p-widths for real projective plane.
method Standard metric used to compute p-widths.
result Computed p-widths for real projective plane.
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
Paper generalizes Satoh's result on knots and projective planes.
problem Understanding connected sums of knots and projective planes.
method Generalizes Satoh's result to odd natural n twist spun spheres and projective planes. result Connected sum of n-twist spun sphere of a knot and an unknotted projective plane is equivalent to the projective plane. In this paper, we study a family of curves on S2 that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective pla…
The study of gyration stability in projective planes.
problem Whether gyrations of projective planes share the same homotopy type.
method Exploration of gyration stability for complex, quaternionic, and octonionic projective planes.
result Complete description of gyration stability for projective planes up to homotopy.
Space of Zoll Finsler metrics on projective plane deformation retracts to round metric.
problem Understanding the structure of Zoll Finsler metrics on projective planes.
method Geodesic flow deformation and curvature flow.
result Space of Zoll Finsler metrics on projective plane is connected and deformation retracts to the round metric.
Study finds specific complex projective plane hypersurfaces hitting equality in curvature inequality.
problem Determining hypersurfaces in complex projective planes hitting equality in curvature inequality.
method Analyzing non-Hopf hypersurfaces with constant mean curvature in the complex projective plane.
result Identifies specific hypersurfaces achieving equality in a curvature inequality.
Classifies geodesic flows on projective plane with potential field.
problem Classifying geodesic flows on a projective plane with a potential field.
method Liouville classification and calculation of Fomenko--Zieschang invariants.
result All Fomenko--Zieschang invariants of the system are calculated.
Study limits of convex domains in projective plane, proving specific results.
problem Understanding limits of convex domains in projective plane.
method Analyzing sequences of properly convex domains with bounded multiplicity.
result Determined all Hausdorff limit domains after normalization.
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.
A new systolic inequality with a remainder for the real projective plane.
problem Proving a stronger systolic inequality for the real projective plane.
method Developing a new systolic inequality with a remainder term.
result A stronger systolic inequality with a remainder for the real projective plane.
Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. We found a unique 4D plane that can't be simplified.
problem Finding an irreducible embedded projective plane in S4. method Constructing a specific geometric shape in 4D space.
result The constructed projective plane is irreducible.
Characterizes hypersurfaces in complex projective and hyperbolic planes.
problem Geometric characterization of hypersurfaces.
method Geometric characterization and partial classifications.
result Characterizes certain hypersurfaces of cohomogeneity one.
Smooth but not symplectic embeddings of rational balls in complex projective plane found.
problem Finding smooth embeddings of rational balls in complex projective plane that are not symplectic.
method Infinite family of rational homology balls, lattice embedding obstruction from Donaldson's diagonalisation theorem.
result No two examples may be embedded disjointly.
Study shows indecomposable branched coverings over projective plane for surfaces with low Euler characteristic.
problem Decomposability of branched coverings over the projective plane with low Euler characteristic surfaces.
method Analyzing coverings of degree d odd, over the projective plane, with surfaces M having χ(M)≤0. result Realization of indecomposable branched coverings for given data with even defect greater than d. Study counts geodesics on special projective planes, finding a noninteger growth rate.
problem Counting geodesics on specific projective planes with removed points.
method Proved true asymptotic formula for geodesics of length ≤ L, independent of hyperbolic structure.
result Growth exponent is noninteger, contrasting with results for orientable surfaces.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
Classifies 4-manifolds up to connected sum with complex projective planes.
problem Classifying 4-manifolds up to connected sum with complex projective planes.
method Classification based on fundamental group, orientation character, and extension class involving second homotopy group.
result Closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of fundamental group, orientation character, and an extension class involving the second homotopy group.
Study uses isotropic projection to link plane and null curves in Minkowski space.
problem Understanding the geometry of null curves in Minkowski 3-space.
method Utilizes isotropic projection of Laguerre geometry to establish a correspondence.
result Alternative description of plane curves congruent to a given one.
We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of comple…
After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to …
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …
New algebraic fundamental groups identified for fake projective planes.
problem Characterizing algebraic fundamental groups of fake projective planes.
method Analysis of complex conjugate pairs and explicit finite étale covers.
result Forty-six distinct isomorphism classes of algebraic fundamental groups.
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
Unique maximal curve systems found for up to 5 punctures.
problem Finding unique maximal curve systems in punctured projective planes.
method Analyzing mapping class group action on maximal 1-systems of loops. result Maximal 1-system is unique for up to 5 punctures. Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
New parametrization handles sextactic points on closed curves.
problem Parametrizing closed projective plane curves with sextactic points.
method Introducing an additional scalar parameter α to define a 2π-periodic global parametrization.
result The balanced parametrization is unique up to a shift of the parameter and is a global projective invariant.
Researchers confirm a 15-vertex triangulation of the quaternionic projective plane is minimal.
problem Prove the quaternionic projective plane has a minimal triangulation with 15 vertices.
method Implemented Gaifullin's algorithm to compute the first rational Pontryagin class of the combinatorial manifold.
result The 15-vertex triangulation is indeed a minimal triangulation of the quaternionic projective plane.
It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.