arXiv research

New findings on minimal isometric immersions of flat n-tori into spheres.

The study restricts stable minimal immersions in product spaces to specific configurations.

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…

Minimal normal curvature immersions in the unit ball studied.

Study minimal Kähler submanifolds in product of space forms.

We prove existence results that give information about the space of minimal immersions of 2-tori into $S ^ 3$. More specifically, we show that \begin{enumerate} \item For every positive integer $n$, there are countably many real $n$-dimensional families of minimally immersed 2-tori in $S ^ 3$. Every linearly ful…

Study of minimal immersions from a sphere to a complex hyperquadric.

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that …

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions …

Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.

Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.

We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…

We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we …

Minimal surfaces can be mapped to 3D with bounded images.

Minimal surfaces can't have certain epitrochoid geodesics.

Classifies minimal immersions from $S^2$ into specific flag manifolds.

We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(σ, tα)$, where $σ$ is a conformal structure on a topological surface $S$, and $αdz^2$ is a holomorphic quadratic differential on the surface $(S,σ)$. We show that, for each $t \in (0,τ_0)$ for some $τ_0…

Constructs minimal immersions with singularities.

The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.

The study proves a strong parametric h-principle for minimal surfaces.

We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…

Minimal hypersurfaces in Q^4(c) cannot be isometrically immersed in Q^4(≠c).

For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.

Paper studies equatorial concentration of measure in sphere immersions and submersions.

In this note, we give natural extensions to cylinders and tori of a classical result due to T. Takahashi about minimal immersions into spheres. More precisely, we deal with Euclidean isometric immersions whose projections in R^N satisfy a spectral condition of their Laplacian.

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a …

In this paper we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above $16 π$. Additionnally we prove an inequality on the second residue for limits sequences of Willmore immersions with simple minimal bubbles. Doing so,…

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

Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $S^2$ …

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

The paper finds new eigenfunctions for minimal immersions and their stability index.

Every nonflat conformal minimal surface is homotopic to a proper one.

We show that any minimal torus in $S^3$ which is Alexandrov immersed must be rotationally symmetric. An analogous result holds for surfaces of constant mean curvature.

The paper extends Weierstrass representation to non-minimal conformal immersions.

Minimal surfaces' area bounds proven equivalent, extending known results.

In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivière's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.

Constructs continuous families of minimal surfaces and holomorphic immersions.

We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.

The paper proves conditions for closed minimally immersed hypersurfaces in a 5-sphere.

Optimal regularity theory for stable minimal hypersurfaces with small singular set.

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally geodesic …

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…

The paper explores generic properties of minimal surfaces in high dimensions.

Study on unique minimal hypersurfaces in rotational domains.

Gauss map of complete minimal surfaces avoids certain hypersurfaces.

New minimal surfaces grow area very quickly.