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.
Paper generalizes splitting number to classify plane curve arrangements.
problem Distinguishing embedded topology of plane curves.
method Introduces splitting graph to generalize splitting number.
result Classifies embedded topology of specific plane curve arrangements.
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.
Study local and global aspects of complex plane curve embeddings.
problem Local and global problems of complex plane curve embeddings.
method Braid monodromy, local and global analysis.
result Historical progress in understanding complex plane curve embeddings.
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.
Every point on an asymptotically flat 3D space has a minimal plane nearby.
problem Finding minimal surfaces in asymptotically flat 3D spaces.
method Proving the existence of minimal planes for every point in the manifold.
result Every point in an asymptotically flat 3D space has a complete properly embedded minimal plane.
The paper proves the existence of minimal planes in specific 3-manifolds.
problem Existence of minimal planes in asymptotically flat 3-manifolds.
method Improving a previous result, the authors prove the existence of minimal planes under specific conditions.
result Properly embedded minimal planes can be found in asymptotically flat 3-manifolds.
The study extends flat plane embedding results to spaces with convex geodesic bicombings.
problem Embedding flat planes in spaces with non-unique geodesics.
method Convexity assumptions on distance functions along geodesics.
result Results on flat plane embedding remain valid in spaces with convex geodesic bicombings.
The paper explores embedded quarters in hyperbolic geometry and limits of ends.
problem Understanding limits of embedded quarters and ends in hyperbolic geometry.
method Proved the existence of sequences of embedded quarters converging to hyperbolic plane ends.
result No algorithm exists to determine if two ends are equal.
We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.
Embedded H-planes found for any curve in hyperbolic 3-space.
problem Finding embedded H-planes for curves in hyperbolic 3-space. method Constructing H-planes with asymptotic boundary curves. result Existence and uniqueness of minimizing H-planes. The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
The study bounds entropy of plane curves and applies to curve shortening flow.
problem Entropy bounds for plane curves and dynamics of CSF.
method Proving entropy lower and upper bounds, constructing curves.
result Entropy of curves is tightly bounded and applied to CSF.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
Study of curves in hyperbolic plane with variable curvature.
problem Finding curves with prescribed almost constant curvature in hyperbolic plane.
method Analyzing closed and embedded curves with geodesic curvature.
result Existence of curves with specified curvature variations.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
Study smooth embeddings of line configurations in complex projective plane.
problem Realizing line configurations as smooth 2-spheres.
method Lattice-theoretic arguments based on Donaldson's diagonalization theorem.
result Established a stronger obstruction in the smooth category.
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.
We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximab…
Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.
problem Understanding exotic smooth embeddings of surfaces in 4-manifolds.
method Analyzing smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli.
result Exotic planes and annuli exhibit radically different properties, with one class being simple enough to draw explicit level diagrams.
In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conj…
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
The paper explores winding numbers of almost embeddings of a 4-vertex graph in the plane.
problem Understanding the winding numbers of almost embeddings of a 4-vertex graph in the plane.
method Constructing examples to show the only relation between the winding numbers of cycles in the graph.
result The sum of winding numbers is odd, and this is the only relation between them.
The paper explores invariants of graph drawings in the plane.
problem Understanding the invariants of almost embeddings of graphs in the plane.
method Proves relations between invariants, connects to homology, constructs examples.
result Constructs almost embeddings realizing some values of invariants.
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedd…
New combinatorial type helps distinguish plane curve topologies.
problem Distinguishing the topology of plane curves.
method Introducing G-combinatorial type using modified plumbing graphs.
result Invariant of G-combinatorial type under certain homeomorphisms.
New symplectic barriers found in ball embeddings.
problem Existence of symplectic embeddings with intersections.
method Proving obligatory intersections with symplectic planes.
result Existence of symplectic barriers in ball embeddings.
In the recent paper \cite{DGNP} we have proved that the only stable C2 minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this pap…
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.
The paper explores how configurations of lines can be realized in different geometric settings.
problem Whether configurations of lines can be realized by spheres in various geometric settings.
method Investigates realizability of configurations in topological, symplectic, and smooth categories in the complex projective plane.
result Obstructions to realizability in the topological category for configurations specified by projective planes over finite fields.
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar graphs.
The plane and catenoid are the only capillary minimal surfaces outside a unit ball with one end and finite total curvature.
problem Characterizing capillary minimal surfaces in 3D space.
method Analytical proof using geometric properties and curvature constraints.
result The plane and catenoid are the only capillary minimal surfaces under specified conditions.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…
The family of embedded, singly periodic minimal surfaces of Riemann have as limit-surfaces the helicoid, the catenoid, a single plane, or an infinite set of equally-spaced parallel planes.
Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
Study Markov staircases in symplectic embeddings of rational homology ellipsoids.
problem Symplectic embeddings of rational homology ellipsoids into the complex projective plane.
method Analysis of almost toric fibrations and Hamiltonian isotopies.
result Existence of an infinite staircase for each Markov triple.
Proves lattice isomorphic arrangements can have equivalent complements but non-homeomorphic embeddings.
problem Exploring topological properties of lattice isomorphic arrangements.
method Proved existence of arrangements with equivalent complements but non-homeomorphic embeddings.
result Existence of lattice isomorphic arrangements with equivalent complements but non-homeomorphic embeddings.
New example of surface flow converging to a plane with multiplicity 2.
problem Constructing mean curvature flows with specific convergence properties.
method Constructing a new example of a mean curvature flow in R3. result The flow converges to a plane with multiplicity 2 as time approaches infinity.
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.
Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.
problem Characterizing limit sets of knots in complex hyperbolic geometry.
method Analyzing embeddings of knots as limit sets of discrete subgroups of PU(2, 1).
result Knots are either chains or R-circles as limit sets.
We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded λ-convex curve of a given length on the Lobachevsky plane.
In hyperbolic 3-space, H-planes can be formed around curves with smooth points.
problem Forming H-planes in hyperbolic 3-space.
method Analyzing Jordan curves in the asymptotic sphere.
result Embedded H-planes can be created for any H in [0,1) around curves with smooth points.
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…
We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to a conjecture of Gabai. Soma has already proven this conjecture earlier. Our tec…
For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.
We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in R3 that are not rotationally symmetric. The strategy for the construction is to start with a family of initi…
Symplectic embeddings of pinwheels into CP² relate to Markov numbers.
problem Embedding pinwheels into the complex projective plane.
method Study Lagrangian embeddings of pinwheels, showing connections to Markov numbers.
result Pinwheels can only embed into CP² if their parameters are Markov numbers.