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 closed braids with the same axis up to orientation. If S is closed without boundary, the contribution of p to the degree of the normal bundle of S in M can be computed on the link L(p) in terms of the algebraic crossing numbers of its components and on their linking numbers one with another.
In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to Hélein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in $\mathbb{R}^{n}$ with uniformly bounded areas and Willmore energies. Furthermore, we prove that the integral identity of Gauss curvature is true.
It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from $\mathbb{S…
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 stationary in the cylinder B \times {\mathbb R}^{k} must be a C^{1, 1/2} function, and the set of its branch points, if non-empty, must have Hausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorff measure. The C^{1, 1/2} regularity is optimal.
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 can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in $R^m$, away possibly from isolated branched points, which minimizes the Willmore …
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.
An equivariant minimal surface in $\mathbb{CH}^n$ is a minimal map of the Poincaré disc into $\mathbb{CH}^n$ which intertwines two actions of the fundamental group of a closed surface $Σ$: a Fuchsian representation on the disc and an irreducible action by isometries on $\mathbb{CH}^n$. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corre…
In the present work we study the behavior of sequences of smooth global isothermic immersions of a given closed surface and having a uniformly bounded total curvature. We prove that, if the conformal class of this sequence is bounded in the Moduli space of the surface, it weakly converges in W^{2,2} away from finitely many points, modulo extraction of a subsequence, to a possibly branched weak isothermic immersion of this surface. Moreover, if this limit happens to be smooth away from the branched points, we give an optimal description of the possible loss of strong compactness of such a subse…
We consider a closed Willmore surface properly immersed in ${\R}^m$ (m>2) with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the "conservative" reformulation of the Willmore equation introduced in a previous paper by the second author, we show that, in an appropriate conformal parametrization, the gradient of the Gauss map of the immersion has bounded mean oscillations if the singularity has order one, and is bounded if the order is at least two. We develop around the singular point local asymptotic expansions for the immersi…
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 characterize the branched double covers $T^2\to S^2 \times \{0\}$ as the only branched conformal immersions, up to Möbius transformations of $\mathbb{R}^4$, from a torus into $\mathbb{R}^4$ with at least one branch point and Willmore energy $8π$. Using a perturbation argument in order t…
In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (possibly branched) maps from $\mathbb{R} P^2$ to $S^4$. The Veronese sphere and several other examples of non-branched Willmore immersions from $\mathbb{R} P^2$ to $S^4$ are derived as an illustration of the general theory. The Willmore immersions of $\mathbb{R} P^2$, just mentio…
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 sequence of times $t_i\to\infty$ 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, w…
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 minimal surface or an inverted Chen's minimal graph must be close to these surfaces in the $W^{2,2}$-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces.
We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $γ$ of $π_2(M^m)$ the limiting lipschitz map…
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 (including min max methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for M^m. To this ai…
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 groups is injective, there exists a branched minimal immersion of $Σ$ solving the free boundary problem $(Σ, \partial Σ) \rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. Furthermore, under certain nonnegativity assu…
We study sequences $f_k:Σ_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq Λ$. Assume that $Σ_k$ converges to $Σ$ in moduli space, i.e. $ϕ_k^\ast(Σ_k) \to Σ$ as complex structures for diffeomorphisms $ϕ_k$. Then we construct a branched conformal immersion $f:Σ\to \R^n$ and Möbius transformations $σ_k$, such that for a subsequence $σ_k \circ f_k \circ ϕ_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $Λ< 8π$ the map $f$ is unbranched. If the $Σ_k$ diverge in moduli space, then we show $\liminf_{k \to \inf…
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 (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with …
We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay in a compact domain of the moduli space.
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 [16,18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed mor…
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 critical points of the Willmore energy or the frame energy for tori are smooth analytic surfaces, away possibly from isolated branched points, under the condition that either the genus is at most 2 or if the sub-manifold does not intersect the subspace of hyper-elliptic points. Using a…
For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is the induced metric is conformally equivalent to the unit Euclidean punctured disk, for any $r\in(0,1)$. For a locally $W^{2,2}$ Lipschitz immersion $f$ from the punctured disk $D_2\backslash\{0\}$ into $\mathbb{R}^n$, if $\|\nabla f\|_{L^\infty}$ is finite and the second fundamen…
A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in $\mathbb{C}^2$ are enough to determine the knot type, we talk of a simple minimal knot. Such a knot is given by three integers $N < p,q$; denoted by $K(N,p,q)$, it can be parametrized in the cylinder as $e^{iθ}\mapsto (e^{Niθ},\sin qθ,\cos pθ)$. From this expression stems a natural representation of $K(N,p,q)$ as an $N$-braid. In this paper…
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 mappings which are not homotopic to a constant mapping, is an immersion. That is, $f$ is free of branch points. As a major step toward treating minimal surfaces of the type of the projective plane, we extend the fundamental theorem of branched immersions to the nonorientable case. …
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $Σ$ 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 li…
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 the branch point, the data of the double points of the $F_t$'s give us a braid representation of this knot as a product of bands.
Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.
We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximizing metrics for the normalized first eigenvalue on any closed surface of fixed topological type. Since these metrics are induced by (possibly branched) minimal immersions into spheres, we find new examples of immersed minimal surfaces in spheres.
We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface $Σ$, possibly with boundary, a closed ambient Riemannian manifold $(\mathcal{M}^m,g)$ and a closed embedded submanifold $\mathcal{N}^n\subset\mathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Φ$ for the functional \begin{align*} &E_σ(Φ):=\operatorname{area}(Φ)+σ\operatorname{length}(Φ|_{\partialΣ})+σ^4\int_Σ|{\mathrm {I\!I}}^Φ|^4\,\operatorname{vol}_Φ\end{align*} on immersions $Φ:Σ\to\mathcal{M}$ with the constraint $Φ(\partialΣ)\subseteq\mathcal{N}$, a…
