Minimal Immersions Research Articles

Complete minimal surfaces densely lying in arbitrary domains of ℝn

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in $\mathbb{R}^n$ $(n\ge 3)$, complex curves in $\mathbb{C}^n$ $(n\ge 2)$, holomorphic null curves in $\mathbb{C}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.

A viscosity method in the min-max theory of minimal surfaces

We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $\Sigma $ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^{q}$ norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition for $q>2$ . This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the “viscosity” parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in Pigati and Riviere ( arXiv:1708.02211 , 2017) that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.

Every Meromorphic Function is the Gauss Map of a Conformal Minimal Surface

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 $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb{R}^n$ for any $n\ge 3$.

AdS 3-manifolds and Higgs bundles

In this paper we investigate the relationships between closed AdS 3 3 -manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for their volume. We give natural foliations of the AdS structure with time-like geodesic circles and we use these circles to construct equivariant minimal immersions of the Poincaré disc into the Grassmannian of time-like 2-planes of R 2 , 2 \mathbb {R}^{2,2} .

On the moduli of isotropic and helical minimal immersions between spheres

On the moduli of isotropic and helical minimal immersions between spheres

Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is $$S^2$$ , we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kahler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kahler angle if it is neither holomorphic nor anti-holomorphic with Kahler angle $$\alpha \ne 0$$ (resp. $$\alpha \ne \pi $$ ) everywhere on $$S^2$$ .

Classification of homogeneous minimal immersions from $$S^2$$ S 2 to $${\mathbb {H}}P^n$$ H P n

In this paper we determine all homogeneous minimal immersions of 2-spheres in quaternionic projective spaces \({\mathbb {H}}P^n\).

Injectivity of minimal immersions and homeomorphic extensions to space

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass–Enneper lifts defined in the unit disk $$\mathbb{D}$$ endowed with a conformal metric. Among the corollaries, we obtain a Becker type condition and a sharp condition depending on the Gaussian curvature and the diameter for an immersed geodesically convex minimal disk in $$\mathbb{R}^3$$ to be embedded. Extremal configurations for the criteria are also determined, and can only occur on a catenoid. For non-extremal configurations, we establish fibrations of space by circles in domain and range that give a geometric analogue of the Ahlfors–Weill extension.

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\mathbb{R}^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a ``constant lightlike vector''. Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in $\mathbb{R}^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.

Almost conformally flat hypersurfaces

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the round sphere and extend a result due to Shiohama and Xu (J. Geom. Anal. 7 (1997) 377–386) for compact hypersurfaces in any space form.

Low energy canonical immersions into hyperbolic manifolds and standard spheres

We consider critical points of the global L2-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of the first of the functionals mentioned.

Complete minimal submanifolds with nullity in Euclidean space

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let \(M^m\) be a complete Riemannian manifold and let \(f:M^m\rightarrow \mathbb {R}^n\) be a minimal isometric immersion with index of relative nullity at least \(m-2\) at any point. We show that if the Omori–Yau maximum principle for the Laplacian holds on \(M^m\), for instance, if the scalar curvature of \(M^m\) does not decrease to \(-\infty \) too fast or if the immersion f is proper, then the submanifold must be a cylinder over a minimal surface.

Classification of conformal minimal immersions of constant curvature from $$S^2$$ S 2 to $$Q_n$$ Q n

In this paper, we investigate geometry of conformal minimal two-spheres immersed in $$G(2,n;\mathbb {R})$$ and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $$S^2$$ to $$G(2,n;\mathbb {R})$$ , or equivalently, a complex hyperquadric $$Q_{n-2}$$ under some conditions.

Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening flow and minimal Legendrian immersions in odd-dimensional spheres. As a consequence, we classify the rotationally invariant Lagrangian homothetic solitons for the inverse mean curvature flow.

Refined asymptotics of the Teichmüller harmonic map flow into general targets.

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 tge 0, we find a sequence of times t_irightarrow infty at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as irightarrow infty .

A rapid and practical technique for real-time monitoring of biomolecular interactions using mechanical responses of macromolecules.

Monitoring biological reactions using the mechanical response of macromolecules is an alternative approach to immunoassays for providing real-time information about the underlying molecular mechanisms. Although force spectroscopy techniques, e.g. AFM and optical tweezers, perform precise molecular measurements at the single molecule level, sophisticated operation prevent their intensive use for systematic biosensing. Exploiting the biomechanical assay concept, we used micro-electro mechanical systems (MEMS) to develop a rapid platform for monitoring bio/chemical interactions of bio macromolecules, e.g. DNA, using their mechanical properties. The MEMS device provided real-time monitoring of reaction dynamics without any surface or molecular modifications. A microfluidic device with a side opening was fabricated for the optimal performance of the MEMS device to operate at the air-liquid interface for performing bioassays in liquid while actuating/sensing in air. The minimal immersion of the MEMS device in the channel provided long-term measurement stability (>10 h). Importantly, the method allowed monitoring effects of multiple solutions on the same macromolecule bundle (demonstrated with DNA bundles) without compromising the reproducibility. We monitored two different types of effects on the mechanical responses of DNA bundles (stiffness and viscous losses) exposed to pH changes (2.1 to 4.8) and different Ag+ concentrations (1 μM to 0.1 M).

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

AbstractWe establish a frequency function monotonicity formula for two‐valued C1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C1, 1/2, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.

Projective Surfaces and Pre-Normalized Blaschke Immersions of Codimension Two

We prove that any non-degenerate surface in the projective 3-space has a local lift as a minimal pre-normalized Blaschke immersion into the equicentroaffine 4-space. Furthermore, an indefinite surface in the projective 3-space has a local lift as a pre-normalized Blaschke immersion into the equicentroaffine 4-space satisfying the Einstein condition if and only if the surface is projectively applicable to an affine sphere.

Classification of conformal minimal immersions of constant curvature from $S^2$ to $Q_3$

In this paper, we study geometry of conformal minimal two-spheres immersed in complex hyperquadric $Q_3$. We firstly use Bahy-El-Dien and Wood's results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from $S^2$ to $G(2,5;\mathbb{R})$. Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from $S^2$ to $G(2,5;\mathbb{R})$, or equivalently, a complex hyperquadric $Q_3$.

Isometric deformations of isotropic surfaces

It was shown by Ramanathan (Manuscripta Math 60:417–422, 1988) that any compact oriented minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. In this paper, we extend this result to isotropic surfaces in spheres of higher dimension. The case of isotropic surfaces other than compact in space forms is also addressed.