Engel structures and flows on 4-manifolds linked via constant rank intersections.
problem Pairs of Engel structures on 4-manifolds with constant rank intersections.
method Established a correspondence with weakly hyperbolic flows.
result Engel structures and flows on 4-manifolds are linked via constant rank intersections.
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.
problem Proving a Minkowski inequality for specific types of hypersurfaces.
method Using weakly mean convex and star-shaped hypersurfaces in warped cylinders, and applying the inverse mean curvature flow.
result Sharp inequality holds for outward minimizing hypersurfaces in Schwarzschild and hyperbolic spaces.
The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
Study of weakly Kähler hyperbolic manifolds, proving Lang and Green-Griffiths conjectures.
problem Verifying Lang and Green-Griffiths conjectures for a new class of manifolds.
method Introducing weakly Kähler hyperbolic manifolds and investigating their spectral properties.
result Proving weakly Kähler hyperbolic manifolds are of general type and verifying various aspects of Lang and Green-Griffiths conjectures.
We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to −1 and are C0, but are not necessarily C1, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves a…
New hyperbolicity concepts expand manifold study.
problem Studying hyperbolicity on complex manifolds.
method Introducing sG-hyperbolicity, weakly p-Kähler hyperbolic structures, and pluriclosed star split hyperbolic metrics.
result Expands the class of divisorially hyperbolic manifolds.
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov'…
Study shows expanding Ricci solitons from specific metric cones.
problem Analyzing Ricci flows from weakly PIC1 metric cones.
method Complete weakly PIC1 Ricci flows with Euclidean volume growth.
result Ricci flows must be expanding gradient Ricci solitons.
Unified definition of mass aspect function for weakly regular hyperbolic manifolds.
problem Ambiguity in mass definition for asymptotically hyperbolic manifolds.
method Introduced an ADM-style mass aspect function for broad asymptotics and low regularity.
result Unified mass aspect function exhibits favorable covariance properties.
The study classifies weakly almost Fuchsian manifolds and proves geometric properties.
problem Classifying and understanding weakly almost Fuchsian manifolds.
method Geometric analysis and compactification techniques.
result Uniform upper bounds on volume and Hausdorff dimension for limit sets.
New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.
problem Characterizing domains without parabolic minimal submanifolds and weakly hyperbolic domains.
method Analyzing properties of domains and their boundaries, using tubular neighborhoods and conformal harmonic maps.
result Domains without parabolic minimal submanifolds and weakly hyperbolic domains have specific geometric properties.
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Q…
Let M be a closed oriented surface endowed with a Riemannian metric g and let Ω be a 2-form. We show that the magnetic flow of the pair (g,Ω) has zero asymptotic Maslov index and zero Liouville action if and only g has constant Gaussian curvature, Ω is a constant multiple of the area form of g and the mag…
We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains have and are more or less determined implicitly by a mean value property for harmonic functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and obtain essen…
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
The study finds knots with specific surgeries that don't allow weak symplectic fillings.
problem Detecting weakly symplectic fillability of L-space knots after positive surgeries. method Analyzing arithmetic data from knot type and surgery coefficients to compute geometric invariants.
result Provides an infinite family of hyperbolic L-spaces that do not admit weakly symplectic fillings. The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3.
Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the co…
Study of polyhedra on a sphere in projective 3-space.
problem Characterizing polyhedra with vertices on a sphere.
method Purely combinatorial and linear programming approaches.
result Characterization of dihedral angles and hyperbolic-de Sitter structure.
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.
Ancient solutions to Ricci flow with isotropic curvature conditions are classified.
problem Classifying ancient solutions to Ricci flow with isotropic curvature conditions.
method Analyzing properties of ancient solutions with isotropic curvature conditions.
result Ancient solutions to Ricci flow with isotropic curvature conditions are either shrinking cylinders or the Bryant soliton.
Classifies surfaces in hyperbolic space with constant Gaussian curvature.
problem Classifying surfaces in hyperbolic space with specific curvature.
method Loop group method, spectral parameter deformation, holomorphic quadratic differentials.
result Weakly complete constant Gaussian curvature surfaces are in one-to-one correspondence with holomorphic quadratic differentials.
In this paper, we find infinite hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that part of these manifolds do admit tight contact structures.
We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem und…
The paper examines the co-rank of weakly parafree 3-manifold groups and provides counterexamples.
problem Investigating the co-rank of weakly parafree 3-manifold groups and their relationship to free groups.
method Constructing specific examples of homology handlebodies and analyzing their fundamental groups.
result The fundamental group of W. Thurston's tripus manifold is not very large, showing that weakly parafree groups of rank at least 3 are not necessarily very large.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal h…
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resembl…
Ancient solutions of Ricci flow with Type I growth are classified.
problem Understanding ancient solutions of Ricci flow with specific curvature growth.
method Analyzing ancient solutions with Type I curvature growth in arbitrary dimensions.
result Ancient solutions with Type I growth are classified into specific types.
New method improves tensor completion for weakly-dependent spatiotemporal data.
problem Improving tensor completion for weakly-dependent data on graphs.
method Introducing L1-norm and Graph Laplacian penalties for low-rank tensor decomposition and completion. result Improved performance in metro passenger flow prediction.
New manifold structure for rel. data, proving new inequalities.
problem Defining a manifold structure for relativistic initial data.
method Adapted Bartnik's method for AH manifolds with inner boundary.
result Proved new weighted Poincaré and Korn inequalities.
A Coxeter n-orbifold is an n-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order m, whose neighborhood is locally modeled on Rn modulo the dihedral group of order 2m generated by two reflections. For n≥3, we study…
We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenome…
Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
problem Characterizing Anosov flows in hyperbolic 3-manifolds.
method Analyzing the properties of Anosov flows and foliations.
result Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
Global existence of Yamabe flows on hyperbolic space proved without curvature bounds.
problem Global existence of Yamabe flows on hyperbolic space without completeness or curvature bounds.
method Instantaneously complete initial metrics, no curvature bounds required.
result Global existence of Yamabe flows on hyperbolic space of arbitrary dimension m≥3. We consider the evolution of hypersurfaces on the unit sphere Sn+1 by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the …
New research finds infinitely many hyperbolic knots not almost-fibered.
problem Understanding the properties of knots and their complements.
method Examined hyperbolic knots and their Seifert surfaces.
result Existence of infinitely many hyperbolic genus one knots that are not almost-fibered.
Acylindrical hyperbolicity proven for a specific group of automorphisms.
problem Proving the acylindrical hyperbolicity of a group of automorphisms.
method Study of a 2D simplicial complex and proving its contractibility and Gromov-hyperbolicity; finding loxodromic elements.
result Tame automorphism group is acylindrically hyperbolic.
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
Paper studies flow on hyperbolic surfaces to match boundary lengths.
problem Matching boundary lengths of hyperbolic surfaces.
method Combinatorial Yamabe flow on hyperbolic bordered surfaces.
result Flow converges exponentially to a surface with equal boundary lengths.
This paper proves energy convexity for bi-harmonic maps into spheres, with applications to heat flow and uniqueness.
problem Analyzing the geometric properties of bi-harmonic maps and their heat flow.
method Energy convexity and ε-regularity of bi-harmonic maps.
result Uniqueness of weakly intrinsic bi-harmonic maps and long-time existence of the heat flow.