Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.
The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
problem The minimality of compact-open topology on diffeomorphism and homeomorphism groups.
method Analyzing the compact-open topology on diffeomorphism and homeomorphism groups of smooth manifolds.
result The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
New stable minimal hypersurfaces found in 4-manifolds, proving topology results.
problem Finding stable minimal hypersurfaces with specific topologies in 4-manifolds.
method Geometric measure theory and 4-manifold topology techniques.
result Existence of stable minimal hypersurfaces diffeomorphic to S3 or S2imesS1. We show that except for n=2 if a bridge surface for a knot is an index n topologically minimal surface, then after a perturbation it is still topologically minimal with index at most n+1.
Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
Minimal surfaces match symmetries and topology exactly.
problem Matching symmetries and topology in minimal surfaces.
method Proved congruence of surfaces with specific symmetries and topology.
result Lawson surfaces are uniquely determined by their symmetries and topology.
Lower bounds on cone density for nontrivial complements in low dimensions.
problem Finding density limits for minimal cones with nontrivial complements.
method Proving lower bounds on cone density for cones of dimensions less than seven with nontrivial complements.
result Established lower bounds on cone density for minimal cones with nontrivial complements in dimensions less than seven.
We give a complete topological classification of minimal surfaces in Euclidian three-space.
We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
We show that an (n+1)-bridge sphere for the unknot is a topologically minimal surface of index at most n.
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.
problem Understanding index growth of minimal hypersurfaces.
method Partitioning methods for compact Lie groups, applied to hypersurfaces.
result Linear index growth for hypersurfaces of fixed topological type.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
In 1997, Collin proved that any properly embedded minimal surface in R3 with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb…
Paper proves timelike minimal surfaces with any number of ends exist.
problem Existence of timelike minimal surfaces with arbitrary ends.
method Analyzes asymptotic behavior and topology of constructed surfaces.
result Timelike minimal surfaces with arbitrary ends exist.
Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.
problem Determining minimal surfaces from boundary data.
method Developed a semiclassical nonlinear calculus for complex geometric optics solutions.
result Minimal surfaces can be recovered from the Dirichlet-to-Neumann map under certain conditions.
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…
Study shows certain toric arrangements have minimal topological complements.
problem Understanding the topology of toric arrangements.
method Associated a matrix to toric arrangements and analyzed those with maximal rank.
result Complement manifolds of these toric arrangements are diffeomorphic to centered ones.
The study proves that certain minimal hypersurfaces in 4D space must be planes.
problem The extension of the half-space theorem to higher dimensions is obstructed.
method Analyzes topological properties of minimal hypersurfaces in R4. result Complete, properly embedded minimal hypersurfaces in R4 with bounded curvature and diffeomorphic to R3 must be planes. The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.
Minimal hypersurfaces in spheres generated by isoparametric foliations are found.
problem Existence of minimal hypersurfaces in spheres generated by isoparametric foliations.
method Generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, reducing the minimal surface equation to an ordinary differential equation.
result Closed embedded minimal hypersurfaces of topological type S1imesM are found for any isoparametric hypersurface M⊂Sn. A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.
problem Extending the half-space theorem to higher dimensions in R4. method Analyzing the topological constraints on minimal hypersurfaces in R4. result A complete, properly embedded minimal hypersurface in a slab in R4 must be a hyperplane. In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.
In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M, an open domain M⊂M with the fixed topological type, and a conformal complete minimal immersion X:M→R3 which can be extended to a continuous map X:Mˉ→R3, such that $X_{|\partial M…
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
We show that a topologically minimal disk in a tetrahedron with index n is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.
New minimal hypersurfaces in 4D sphere found.
problem Constructing embedded minimal hypersurfaces in S4. method Equivariant min-max theory and suspended Hopf action.
result Infinitely many topological S1-bundles and Seifert fibered manifolds found. Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
Study minimal freezing sets in convex digital disks.
problem Finding minimal freezing sets in convex digital disks.
method Showed how to find minimal freezing sets for convex disks in digital plane.
result Found minimal freezing sets for convex disks in digital plane.
We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3-manifold N. We also obtain a least area, incompressible, properly embedded, finite topology, 2-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…
Study on stellar models' topology and mass using minimal surfaces.
problem Investigating the topology and mass of static stellar models.
method Analyzing stable free boundary minimal surfaces in static perfect fluid spaces.
result Proved non-existence of stable free boundary minimal surfaces and derived upper bounds for Hawking mass.
We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil3 using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal surfaces in Nil3 with non-trivial topology. Moreover, we will classif…
Introduces topological deep learning for neural network classification problems.
problem Classifying neural networks using minimal topological structures.
method Formalizes classification problems in a topological setting.
result Demonstrates conditions for the feasibility of classification problems in neural networks.
New EZ-structure maps mapping class group actions.
problem Mapping class group actions on surfaces.
method Constructing a boundary with minimal, strongly proximal, and topologically free action.
result Boundary acts on mapping class group with desired properties.
Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the Riemannian metric.
The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
We consider two natural classes of minimal laminations in three-manifolds. Both classes may be thought of as limits - in different senses - of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations are topologically either disks,…
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.