arXiv research

Odd co-dimension Riemannian foliations don't work on curved surfaces.

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.

Study co-dimension one area-minimizing currents with tangentially immersed boundaries and co-oriented mean curvature.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

We consider the timelike minimal surface problem in Minkowski spacetimes and show local and global existence of such surfaces having arbitrary dimension $\geq 2$ and arbitrary co-dimension, provided they are initially close to a flat plane.

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.

We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $…

Proves local existence and uniqueness of SMCF in Euclidean spaces.

The study explores embedding closed contact manifolds in higher dimensions.

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

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.

We show that every analytic semi-Riemannian manifold can be isometrically embeddded into an Einstein maifold in co-dimension one.

Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.

Partial boundary regularity for area-minimizing currents at tangential boundary points.

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…

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.

We study knots in 3d Chern-Simons theory with complex gauge group $SL(N,\mathbb{C})$, in the context of its relation with 3d $\mathcal{N}=2$ theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d $(2,0)$ theory, which is compactified on a 3-manifold $\hat{M}$. …

The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.

Curved surfaces shrink to points via mean curvature flow.

The study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.

New formulation tackles arbitrage in volatile markets using eigenvalue bounds.

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 umbilical properties of spacelike 2D submanifolds in semi-Riemannian geometry.

Study investigates multicomponent bilayers in Cahn-Hilliard models.

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 $S^5$ 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 $S^{n}$ by reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a special example. I…

The paper finds conditions for biharmonic orbits in symmetric spaces.

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

The paper constructs a Hitchin connection for a broad class of Kähler structures.

Extends index theorem to domain walls with discontinuous Riemannian connections.

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to …

The paper proves conditions for positivity preservation on Riemannian manifolds.

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 …

Paper equates torsions on wedge singularities.

Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then …

Study on special Kähler metrics with specific asymptotic properties.

In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is exactly the cubic form. We also consider parameterizations on two dimensional affi…

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the…

The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.

Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.

The index theorem connects anomalies on a domain wall to global integrals.

Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1,1) tensor…

The paper studies variations of metrics on foliated manifolds and finds solutions to specific actions.