arXiv research

Sharp bounds on singularities of stable minimal hypersurfaces.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

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

Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.

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

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those ma…

The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.

Optimal regularity for 2D stationary varifolds proved.

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a …

The paper generalizes a theorem about metrics realizing maxima of the first non-zero Laplace eigenvalue.

Proves existence and regularity of spherical minimizers for lipid membrane energy.

The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (includin…

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 …

Study on membranes under confinement, proving existence and regularity of minimizers.

Maximal metric found for first Steklov eigenvalue on surfaces.

Proof confirms minimal surfaces have single instance in arbitrary dimensions.

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minim…

Method calculates Morse index of branched Willmore spheres in 3-space.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

Study investigates uniform convergence of Teichmueller harmonic map flow.

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (…

The study examines the regularity of branched immersions using special coordinate systems.

The paper extends a theorem and studies the convergence of branched conformal immersions with bounded areas and Willmore energies.

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key…

New metrics found by attaching cylinders or cross caps to surfaces.

Researchers define and prove existence of minimizers for generalized Willmore functionals.

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…

Stability of branched immersions with energy constraints.

The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [1…

Study on framed surfaces with bounds on Morse index.

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…

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

Proves existence of non-planar minimal disks in ellipsoids.

The paper analyzes cyclic Higgs bundles using elliptic systems and immersion properties.

Flow maps into minimal surfaces with free boundary.

LCD n-manifolds are linked to branched n-manifolds.

For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4πk$. Moreover, we show that the energy value $8π$ cannot be attained by such an immersion. Additionally, we charact…

The paper details folding of branched covers of the 3-sphere over knots.

MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $Σ$ be a compact orientable surface with boundary. We show that for any continuous $f: (Σ, \partial Σ) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental gro…

This paper studies special Lagrangian submanifolds and their deformations.

Study $d$-invariants of L-space double branched covers of arborescent links.

New method shows stability of Willmore immersions' Morse index and nullity.