Odd co-dimension Riemannian foliations don't work on curved surfaces.
problem Proving Riemannian foliations can't exist on positively curved manifolds.
method Reasonable conditions on co-dimension and curvature.
result Odd co-dimension Riemannian foliations cannot occur in positively curved manifolds.
We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.
Defines conditions for umbilical submanifolds in arbitrary dimensions.
problem Conditions for umbilical submanifolds in arbitrary dimensions.
method Using the total shear tensor and defining shear and umbilical spaces.
result The sum of dimensions of shear and umbilical spaces equals the co-dimension.
Study co-dimension one area-minimizing currents with tangentially immersed boundaries and co-oriented mean curvature.
problem Understanding area-minimizing currents with specific boundary conditions.
method Introducing and studying co-dimension one area-minimizing currents with tangentially immersed boundaries and co-oriented mean curvature.
result Any such currents are supported in a smooth hypersurface near the boundary, with tangent cones being hyperplanes of constant orientation but non-constant multiplicity.
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
We consider the timelike minimal surface problem in Minkowski spacetimes and show local and global existence of such surfaces having arbitrary dimension ≥2 and arbitrary co-dimension, provided they are initially close to a flat plane.
Study smooth mean curvature flow for arbitrary co-dimension sets with weak tangent planes.
problem Existence and uniqueness of smooth mean curvature flow from Reifenberg flat sets.
method Viscosity solutions and level set approach for arbitrary co-dimension.
result Non-fattening and smooth flow for small Reifenberg parameter.
We make several improvements on the results of M.-T. Wang in [8] and his joint paper with M.-P. Tsui [7] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature condition and lower bound of ∗Ω are weakened. New applications are also obtained.
A new sub-bundle structure on exotic and standard spheres proven.
problem Constructing a co-dimension 3 sub-bundle on exotic and standard spheres.
method Using Sp(2)-principal bundles and Hopf bundles, the method provides an alternate proof.
result A co-dimension 3 sub-bundle on the Gromoll-Meyer exotic 7-sphere and standard 7-sphere.
Proves local existence and uniqueness of SMCF in Euclidean spaces.
problem Proving existence and uniqueness of SMCF.
method Local existence and uniqueness proof for SMCF.
result Local existence and uniqueness of SMCF proved.
The study explores embedding closed contact manifolds in higher dimensions.
problem Iso-contact embeddings of closed contact manifolds in co-dimension 2.
method Analyzes conditions for iso-contact embeddings and applies results to specific cases.
result Closed contact 3-manifolds without 2-torsion in their cohomology iso-contact embed in the standard contact 5-sphere.
Embeddings in 5D can be isotoped to closed braids.
problem Embeddings of manifolds in high dimensions.
method Piecewise linear co-dimension two embeddings, isotopy to closed braids.
result Embeddings in 5D can be isotoped to closed braids.
We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we …
We disproving Seifert's conjecture for almost symplectic foliations with co-dimension bigger or equal to 3.
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
We show that every analytic semi-Riemannian manifold can be isometrically embeddded into an Einstein maifold in co-dimension one.
We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …
Partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents.
method Proof closely follows Hardt and Simon's boundary regularity result.
result Partial regularity of tangent cones, uniqueness of tangent cone.
We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which ha…
Study shows partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents at tangential points.
method Proof follows Hardt and Simon's boundary regularity result in [9], focusing on currents with tangent cones in hyperplanes.
result Partial regularity and uniqueness of tangent cones for currents with tangential boundary.
We give explicit representation formulas for marginally trapped submanifolds of co-dimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature. This paper is dedicated to the memory of Franki Dillen, 1963-2013.
The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.
problem Proving uniqueness of blow-down for critical points of a Yang-Mills-Higgs functional.
method Using an Allard-type improvement of flatness to establish co-dimension-two analogue of Savin's theorem.
result Entire critical points have unique blow-down, two-dimensional in ambient dimensions 2-4 or any dimension assuming local minimizer.
Curved surfaces shrink to points via mean curvature flow.
problem Curvature pinched surfaces.
method Mean curvature flow.
result Surfaces shrink to round points.
The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.
problem Existence of small volume hyperbolic 4-manifolds with embedded 3-manifolds.
method Analysis of hyperbolic manifolds and their submanifolds.
result Minimal volume hyperbolic 4-manifolds with embedded 3-manifolds exist.
This paper examines knots in 3d Chern-Simons theory and their relation to 3d and 5d theories.
problem Understanding knots in 3d Chern-Simons theory and their correspondence with other 3d and 5d theories.
method Analyzing defects in 6d (2,0) theory compactified on a 3-manifold, computing partition functions.
result Identification and quantitative checks of defects in various 3d and 5d theories.
New formulation tackles arbitrage in volatile markets using eigenvalue bounds.
problem Arbitrage opportunities in volatile markets beyond a certain time horizon.
method Formulated as a stochastic optimal control problem, solved via PDE.
result Characterized arbitrage time horizon through PDE solution.
Study umbilical properties of spacelike 2D submanifolds in semi-Riemannian geometry.
problem Characterize umbilical properties of spacelike 2D submanifolds.
method Introduce total shear tensor and shear operators; analyze relationships; consider novel umbilical notions; prove necessary and sufficient conditions for umbilical submanifolds.
result Unique umbilical direction exists unless submanifold is totally umbilical.
We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-man…
Study investigates multicomponent bilayers in Cahn-Hilliard models.
problem Stability and evolution of multicomponent bilayer interfaces.
method Developed mFCH model to study gradient flows of bilayer interfaces, derived sharp interface energy.
result Identified codimensional bifurcation mechanism leading to layer-by-layer pearling.
We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-d…
Ejiri's torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in Sn by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. I…
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
The paper finds conditions for biharmonic orbits in symmetric spaces.
problem Conditions for biharmonic orbits in symmetric spaces.
method Analyzes isotropy representations and biharmonic submanifolds in hyperspheres.
result Necessary and sufficient conditions for biharmonic orbits in symmetric spaces.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
The paper constructs a Hitchin connection for a broad class of Kähler structures.
problem Developing a Hitchin connection for a large class of Kähler structures.
method Generalizing previous work, the paper constructs a Hitchin connection on the space of compatible complex structures on arbitrary symplectic manifolds.
result The constructed Hitchin connection is a partial connection on the tangent space of compatible complex structures, defined in a neighborhood of natural families of complex structures.
Extends index theorem to domain walls with discontinuous Riemannian connections.
problem Index theorem for domain walls with discontinuous Yang-Mills and Riemannian connections.
method Extension of index theorem to new conditions.
result Validates index theorem for more complex discontinuities.
We prove the developability and C1,1/2 regularity of W2,2 isometric immersions of n-dimensional domains into Rn+1. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the W2,2 strong norm, provided the domain is C1 and convex. Both results fail to …
The paper proves conditions for positivity preservation on Riemannian manifolds.
problem Conditions for positivity preservation on Riemannian manifolds.
method Smooth monotonic approximation, subharmonic distributions, and manifold versions of inequalities.
result Conditions for Lp-Positivity Preservation on Riemannian manifolds. The paper defines odd symplectic structures and compares them to Riemannian structures.
problem Defining and comparing odd symplectic structures on supermanifolds.
method Analyzing the Cartan prolongation of Lie algebras to describe differences between odd Riemannian and odd symplectic structures.
result Formulated an analogue of the Levi-Civita theorem for odd symplectic supermanifolds.
We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a lower bound on the number of quadrilaterals of normal surfaces depending on the …
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
Transforms odd-dimensional hyperbolic spaces using parallel formulas.
problem Developing formulas for odd-dimensional hyperbolic spaces.
method Parallel formulas to odd-dimensional spheres.
result Isometry and inversion formulas for Segal--Bargmann transform.
The paper provides a sliceness criterion for stably odd knots.
problem Sliceness of stably odd knots
method Analyzes cobordisms and free knots to derive a sliceness criterion
result A sliceness criterion for stably odd knots
Paper equates torsions on wedge singularities.
problem Analyzing spaces with wedge singularities.
method Equating analytic and intersection Reidemeister torsions.
result Equation of torsions in specific wedge singularities.
Study rigidity theorems for odd dimensional manifolds using Toeplitz operators.
problem Rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds.
method Combining modular method, modular transgression, and analysis of odd Chern classes.
result Fundamental group of manifolds is crucial in odd dimensions for rigidity.
Proves conditions for odd pretzel knots to be doubly slice.
problem Identifying conditions for odd pretzel knots to be doubly slice.
method Analyzes knots with specific twist parameters and odd integers.
result Identifies all doubly slice odd pretzel knots.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…