arXiv research

Paper shows convergence types match for almost minimal sets.

Unique tangent cones found for boundary points of 2D almost-minimizing currents.

Study on biharmonic almost complex structures on compact manifolds.

Study on energy-minimizing structures in complex geometry.

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

The paper analyzes minimal G-structures induced by the Lee form.

The study classifies natural almost Hermitian structures on specific Lie groups.

Study shows almost minimizing rectifiable chains in Hilbert space have regular points dense in their support.

Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

Study soap films hanging from frames, proving surface limits and curvature conditions.

Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.

The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.

New examples show flat singular sets can be arbitrarily complex.

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…

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

Minimal surfaces can have tiny undulations.

Boundary bubbles form in almost minimal cylinders under specific conditions.

Improved optimal regularity for harmonic almost complex structures.

New epiperimetric inequality for cones with improved regularity results.

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…

We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Rie…

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensiona…

We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.

Study horospheres in hyperbolic 3-manifolds with degenerate ends.

We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional curren…

Paper classifies $CFK^\infty$ type of almost L-space knots.

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…

We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents…

New 5-manifold found with zero Ricci curvature.

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one ca…

In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is "almost graph…

Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class $W^{1,p}$, $1 < p < \infty$. If $f$ almost minimizes its $p$ Dirichlet energy then $f$ is Hölder continuous. If $p=2$ and $f$ is squeeze and squash stationary then $f$ is in VMO.

In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers …

The paper extends toric variety correspondence to 4D almost complex torus manifolds.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

Paper proves minimality of stable submanifolds in flat neighbourhoods.

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.

The study shows conditions for almost complex structures on rational homology spheres and limits on their Betti numbers.

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…

The paper classifies almost Yamabe solitons on hypersurfaces and submanifolds in Euclidean spaces.

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by…

Paper explores cohomology classes on non-compact almost Kähler manifolds.

Study of new link types and their invariants, extending previous results.

A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to character…

The paper examines the geometry of specific submanifolds in flag manifolds.

In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.