Extends mean curvature flow results to curved spaces using entropy.
problem Generalizing mean curvature flow results to curved ambient spaces.
method Weighted monotonicity formula and entropy continuity study.
result Derives almost monotonicity for entropy in curved spaces.
Generalizes embedding formalism for CFTs on curved backgrounds.
problem Capturing CFTs on curved backgrounds and non-trivial states.
method Using ambient metric and geometric invariants of the ambient space.
result Exact agreement with holographic computations and thermal OPEs.
Study complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
problem Complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
method Mix of holomorphic curve techniques and convexity results.
result Minimal Lagrangians do not admit fillings by holomorphic discs in negative curvature case.
Extends submanifold theorem to general spaces.
problem Tackles submanifolds in general ambient spaces.
method Uses development of curves in positive codimension and generalizes Cartan-Ambrose-Hicks theorem.
result Provides a geometric construction of isometric immersions.
Curve shortening flow shrinks curves to points under certain conditions.
problem Understanding how curves shrink under curve shortening flow with ambient forces.
method Rescaling and curvature bounds analysis following Gage and Hamilton.
result Curves shrink to round points under certain curvature conditions.
The paper proves new Harnack inequalities for curvature flows in curved spaces.
problem Proving new inequalities for curvature flows in curved spaces.
method Differential Harnack inequalities for flows of strictly convex hypersurfaces by powers of mean curvature in Einstein manifolds.
result New Harnack inequalities for curvature flows in Einstein manifolds with positive sectional curvature.
Refined estimates for surfaces in curved spaces based on Willmore functional.
problem Estimating the position of surfaces in curved spaces accurately.
method Critical points of the Willmore functional, constrained area, refined geometric center of mass.
result Improved position estimates related to ambient scalar curvature.
New spherical curve deformations solve a conjecture.
problem Solving the Östlund Conjecture for spherical curves.
method Introducing a new type of deformation (β) and proving equivalence under specific deformations.
result Equivalence of spherical curves under specific deformations.
The paper extends Euler's problem to hyperbolic and spherical planes.
problem Extending Euler's problem to hyperbolic and spherical planes.
method Characterizing critical points of moment of inertia energy in hyperbolic and spherical planes.
result Closed stationary curves in hyperbolic plane are circles centered at N.
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphis…
T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latt…
Uniform convergence of isotopies implies ambient isotopy, aiding knot equivalence.
problem Determining when uniform convergence of isotopies leads to ambient isotopies.
method Using a diagrammatic condition to offload uniform convergence, constructing examples of tame knots.
result Constructing tame knots with countably-many crossings, distinguishing them from wild curves.
The study classifies totally umbilical surfaces in warped product manifolds.
problem Classifying totally umbilical surfaces in warped product manifolds.
method Analyzing surfaces in warped product M(κ)fimesI and finding first integrals. result Totally umbilical surfaces in M(κ)fimesI are invariant by isometries. Study theta-curves on torus in 3-sphere, classifying them.
problem Classify theta-curves on torus in 3-sphere.
method Analyze nontrivial knots and essential arcs, compute constituent knots, identify structure.
result Complete classification of torus theta-curves up to isotopy and homeomorphism.
We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.
Study geometric properties of RM vector fields in Euclidean, Hyperbolic, and Kähler spaces.
problem Geometric properties of RM vector fields not fully explored.
method Analysis of RM vector fields along curves in Riemannian manifolds.
result Many geometric properties of RM vector fields are studied in specific Riemannian manifolds.
Study curves in a spacelike hypersurface using geometric and singularity theory.
problem Investigate the geometry of curves in spacelike hypersurfaces.
method Use techniques of the theory of singularities to describe the shapes and singularities of hyperbolic and de Sitter surfaces.
result Contribute to the study of extrinsic geometry of curves in different ambient spaces.
This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.
problem Understanding the relationship between Yamada polynomial and Jones polynomial for θ-curves.
method Investigates the equivalence between the normalized Yamada polynomial of θ-curves and the Jones polynomial of their associated links.
result Shows that the two polynomials are equivalent for brunnian θ-curves.
Classifies curved bidifferential operators on manifolds.
problem Classify bidifferential operators on curved spaces.
method Complete classification using Laplacian and conformal invariance.
result Constructs a large class of conformally invariant differential operators.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
Symmetry groups help define solitons in curved spaces.
problem Understanding solitons in curved spaces.
method Defined generalized solitons using symmetry groups.
result Affine solutions are self-similar.
Derives Ribaucour coordinates for curves and submanifolds, smoothing curvature line nets.
problem Deriving Ribaucour coordinates for curves and submanifolds.
method Uses Bianchi permutability result and Ribaucour transformations.
result Reduction of ambient dimension for submanifolds proved.
Geometry-aware noise improves model generalization on complex manifolds.
problem Improving model generalization on highly curved data manifolds.
method Add geometry-aware noise to input space, projecting Gaussian noise onto tangent space of manifold and mapping it via geodesic curve.
result Geometry-aware noise leads to improved generalization and robustness on highly curved manifolds.
Machine learning classifies surface wave dispersion curves from ambient noise.
problem Classifying surface wave dispersion curves from ambient noise.
method Convolutional neural network (U-net) with transfer learning and supervised learning.
result Machine classification nearly identical to human-picked phases.
The paper explores polyharmonic curves in semi-Riemannian manifolds.
problem Investigating polyharmonic curves in semi-Riemannian manifolds.
method Analyzing Frenet curves in semi-Riemannian manifolds of various types.
result Existence, non-existence, and classification results for polyharmonic curves.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
The study defines and characterizes extrinsic catenaries in hyperbolic space.
problem Understanding catenaries in hyperbolic geometry.
method Defined extrinsic catenaries in hyperbolic plane, characterized them, and proved their relation to minimal surfaces.
result Extrinsic catenaries in hyperbolic space are critical points of a potential functional and generating curves of minimal surfaces.
By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a r…
The paper explores projective structures on curves and their applications in conformal geometry.
problem Finding qualitative information about solutions of Hill equations.
method Detailed description of projective structures and their isomorphism classes, correcting previous inaccuracies.
result The Yamabe problem for curves has no general solutions in a conformal/Möbius ambient space.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an amb…
Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.
problem Characterizing and proving the existence of non-CMC biconservative hypersurfaces in spheres.
method Analyzing p-elastic curves of profile curves of biconservative rotational hypersurfaces in space forms. result Existence of a discrete biparametric family of non-CMC closed biconservative hypersurfaces in Sn(ρ), none of which can be embedded. B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We…
New formalism solves kinematical constraints in curved backgrounds and non-trivial states.
problem Solving kinematical constraints due to Weyl invariance in curved backgrounds and non-trivial states.
method Constructing Weyl covariant geometric objects and identifying them as building blocks of correlation functions.
result Exact agreement with thermal OPEs and holographic computations for thermal 2-point functions.
Curve shortening flow increases annulus modulus.
problem Behavior of annulus modulus under curve shortening flow.
method Nested curves evolving under curve shortening flow.
result Modulus of enclosed annulus is monotonically increasing.
Proves transitivity of real Anosov diffeomorphisms with specific properties.
problem Transitivity of real Anosov diffeomorphisms with specific properties.
method Proves transitivity using specific properties of real Anosov diffeomorphisms.
result Proves transitivity of real Anosov diffeomorphisms.
A new method represents rod shapes as paths in special Euclidean algebra.
problem Representing the shapes of rods and framed curves for mechanical analysis.
method Representing shapes as paths in the special Euclidean algebra.
result The method avoids expensive reconstruction and interpolation in rod mechanics.
The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative. High degree Bezier curves are often used as …
Study on k-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.
problem Understanding the growth rate and asymptotic behavior of k-surfaces in negatively curved 3-manifolds. method Proved results on the asymptotic behavior of high energy k-surfaces, including upper bounds and rigidity theorems. result Determined a rigid upper bound for the growth rate of quasi-Fuchsian k-surfaces in negatively curved 3-manifolds. Study of curve evolution in 2D space forms converging to a circle.
problem Understanding curve evolution in 2D space forms.
method Inverse curvature flow with normal speed defined by weighted inverse curvature and support function.
result Solutions exist for all time and converge exponentially to a standard round geodesic circle.
The paper defines catenary curves in spheres and hyperbolic planes.
problem Defining catenary curves in non-Euclidean geometries.
method Characterizations of catenary curves in terms of curvature and angle with geodesics.
result Characterizations and extensions of catenary curves in hyperbolic geometry.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.
problem Understanding geometric properties of curves and surfaces in Riemannian spaces.
method Developing a theoretical framework to study curves and surfaces by their angle with a parallel transported vector field.
result Surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces.
Study projective connections on surfaces using osculating spaces.
problem Understanding projective connections on surfaces.
method Analyzing the second neighborhood and osculating behavior of integral curves.
result Geometrical interpretation of joint invariants of a related group.
Study calculates Gothic Teichmüller curves' Euler characteristics, differing from usual.
problem Computing Euler characteristics of Gothic Teichmüller curves.
method Construction of 'Gothic' Hilbert modular forms.
result Euler characteristic not proportional to ambient surfaces.
I construct "fake algebraic curves" in Cp2. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces F⊂Cp2 homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve a…
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to …