arXiv research

Generalizes embeddedness result for extreme curves.

Study preserves planar and graphical properties of curves under elastic flow.

Smooth convergence shown for curve diffusion flows.

Constructs minimal surfaces by gluing saddle towers with Scherk ends.

Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.

Topology classifies bipolar surfaces; they are not embedded.

This paper has been withdrawn.

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve i…

We find a 2-parameter family of deformations in R^4_1 of the classical Chen-Gackstatter surface explicitly, and show the existence of a larger 4-parameter family of deformations. Each of them still has genus one, a unique end, with total Gaussian curvature $-\int K=8π$. On the other hand, a uniqueness theorem is obtain…

Optimal thresholds ensure curves remain embedded in flows.

Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set…

We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4πmust be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

The flow of symmetric spheres converges to a round sphere.

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proo…

Study constant mean curvature tubes in homogeneous spaces.

Let $X$ denote a metric Lie group diffeomorphic to $\mathbb{R}^3$ that admits an algebraic open book decomposition. In this paper we prove that if $Σ$ is an immersed surface in $X$ whose left invariant Gauss map is a diffeomorphism onto $\mathbb{S}^2$, then $Σ$ is an embedded sphere. As a consequence, we deduce that an…

Study on liquid-vapor interfaces in stable equilibrium without assuming prior regularity.

In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (1995). As a consequence we remove the severe restrict…

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \…

In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$. We prove that for $n \geq 3$ this moduli space is path-connected, and that for $n = 2$ the connected components of the moduli space are in bijective correspondence with the knot…

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

The paper classifies and analyzes the stability of elastic curves with fixed endpoints.

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a l…

Managing large-scale transportation infrastructure projects is difficult due to frequent misinformation about the costs which results in large cost overruns that often threaten the overall project viability. This paper investigates the explanations for cost overruns that are given in the literature. Overall, four categ…

In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic …

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schw…

In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the bou…

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'.…

Survey on conjugate surfaces in product spaces.

Paper proves rigidity for self-similar solutions in 3D flows.

In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic harmonic maps $G : \…

The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.

We prove that if $φ\colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $φ$ is an embedding, and every compact subset of $φ(\mathbb{R}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any pla…

The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.

Study of minimal surfaces in 4D with specific ends.

In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $φ:M^n \to \mathbb{H}^{n+1}$ and a class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some of the key aspects of the correspondence and its consequences have dimensional restrictions $…

Under what conditions is an edge present in a social network at time t likely to decay or persist by some future time t + Delta(t)? Previous research addressing this issue suggests that the network range of the people involved in the edge, the extent to which the edge is embedded in a surrounding structure, and the age…

The paper constructs surfaces with constant mean curvature in a specific space and explores their properties.

Develops explicit formulas for minimal immersions in 5D space.

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold. If the height function of $f$ has a critical point, then it is an embedding and $…