arXiv research

Minimal surfaces and curves can have singularities removed by isotopy.

The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.

New results on hypersurfaces show no branch points, improving smoothness.

Complex wrinkling patterns emerge in non-Euclidean elastic sheets due to energy minimization.

Branch points of a real 2-surface S in a 4-manifold M generalize the branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when S is topologically embedded in M. It defines a link L(p), the components of which are clos…

New estimates show unique cylindrical blow-ups for Dirichlet energy minimizers near singular points.

New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.

Stability of branched immersions with energy constraints.

Analyzes branch points of area-minimizing currents with non-2 planar frequency.

New height estimate for area minimizing currents, leading to unique tangent cones and decay properties.

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves …

Study on singularities of area-minimizing currents, focusing on frequency and branch points.

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…

HADES detects data singularities quickly and accurately.

The paper studies how surfaces move by mean curvature flow and what happens at singular points.

Discrete theory for rational maps improved by generalized branch points.

The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.

In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second …

Paper finds new realizable data for maps with three branch points.

In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in $\R ^ 3$ with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to …

Random free group outer automorphisms are geometric and have nongeometric attracting trees.

Study continuous deformations of branched projective structures on surfaces, preserving holonomy and branch points.

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

Study branched coverings of singular (G,X)-manifolds, solving open questions.

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around …

Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, w…

Closed geodesic nets on surfaces have limited branch points

Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.

We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or …

The paper explores positivity and irreducibility in Hurwitz spaces related to differentials of the second kind.

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projecti…

Machine learning finds Z/2 eigenfunctions on a sphere.

Introduces CSST and characterizes its topology.

The paper explores rational functions with 3 branching points on the Riemann sphere.

Researchers compute specific Hurwitz numbers for branched covers.

For distinct points $p$ and $q$ in a two-dimensional Riemannian manifold, one defines their mediatrix $L_{pq}$ as the set of equidistant points to $p$ and $q$. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes addi…

Proves resurgent nature of a series solution to deformed Painlevé I equation.

Elementary proof shows no specific torus to sphere cover with certain branching points.

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…

We show that any 2-valued C^{1, α} (α\in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2} are distinct, is station…

Character variety of a 6-punctured sphere mapped to CP^3 with specific branch points.

Minimal surfaces in CAT(0) spaces are well-behaved except at a few points.

Rigidity of critical eigensections on spheres proven.

Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.

Researchers compute specific types of branched surface covers.