[Paper Review] $W^{2,2}$-conformal immersions of a closed Riemann surface into $\R^n$

Ernst Kuwert, Yuxiang Li|arXiv (Cornell University)|Jul 22, 2010

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper establishes compactness for sequences of $W^{2,2}$-conformal immersions of closed Riemann surfaces into $ ^n$ with uniformly bounded Willmore energy. It proves that under convergence in moduli space, a subsequence converges weakly in $W^{2,2}_{ m loc}$ to a branched conformal immersion, with unbranched limit if energy is below $8 ilde{ m extbackslash{}pi}$. When moduli space diverges, the energy lower bound is $ ext{min}(8 ilde{ m extbackslash{}pi}, ilde{ m extbackslash{}omega}_p^n)$, generalizing prior results to arbitrary codimension.

ABSTRACT

We study sequences $f_k:Σ_k o \R^n$ 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. $ϕ_k^\ast(Σ_k) o Σ$ as complex structures for diffeomorphisms $ϕ_k$. Then we construct a branched conformal immersion $f:Σ o \R^n$ and Möbius transformations $σ_k$, such that for a subsequence $σ_k \circ f_k \circ ϕ_k o f$ 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 $\liminf_{k o \infty} {\cal W}(f_k) \geq \min(8π,ω^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.

Motivation & Objective

• To establish compactness for sequences of $W^{2,2}$-conformal immersions of closed Riemann surfaces into $ ^n$ with bounded Willmore energy.
• To extend the compactness results of [K-S2] from codimension 3 to arbitrary codimension $n \geq 3$.
• To analyze the behavior of such sequences when the underlying Riemann surfaces diverge in moduli space, establishing a sharp energy lower bound.
• To prove that isolated singularities of $W^{2,2}$ conformal immersions with finite area and $L^2$ second fundamental form are branch points in a weak sense.
• To generalize the Li-Yau inequality and monotonicity formula to the $W^{2,2}$ setting for conformal immersions in higher codimensions.

Proposed method

• Utilizes a Hurwitz-type convergence theorem for conformal immersions, adapted from Hélein’s work, to control weak convergence in $W^{2,2}_{ m loc}$ away from finitely many points.
• Applies the conformal factor estimates from Müller and Šverák to control the growth and degeneration of the immersion near singularities.
• Employs a monotonicity formula from Simon’s work to analyze the concentration of energy near potential branch points.
• Uses the collar lemma and isometric embeddings of degenerating annuli to model neck regions in surfaces with divergent moduli.
• Applies the Li-Yau inequality to control energy lower bounds in the case of multiple bubble tree limits.
• Constructs a blow-up analysis via Möbius transformations and rescaling to identify weak limits and quantify energy concentration.

Experimental results

Research questions

• RQ1 What is the sharp lower bound on the Willmore energy of a sequence of $W^{2,2}$-conformal immersions when the underlying Riemann surfaces diverge in moduli space?
• RQ2 Under what conditions does a sequence of $W^{2,2}$-conformal immersions converge weakly in $W^{2,2}_{ m loc}$ to a branched conformal immersion?
• RQ3 Can the compactness result for Willmore surfaces in $ ^3$ be generalized to arbitrary codimension $n \geq 3$?
• RQ4 What is the role of the conformal factor and its $W^{1,2}$ regularity in controlling the regularity and compactness of conformal immersions?
• RQ5 How do the energy contributions from bubble trees and neck regions behave in higher codimensions, and what is the optimal energy threshold for compactness?

Key findings

• For sequences of $W^{2,2}$-conformal immersions with fixed genus and Willmore energy bounded by $ ilde{ m extbackslash{}Lambda}$, if the Riemann surfaces converge in moduli space, then a subsequence converges weakly in $W^{2,2}_{ m loc}$ to a branched conformal immersion $f: \Sigma \to \r^n$ after Möbius transformations.
• If the Willmore energy is strictly less than $8\tilde{ m extbackslash{}pi}$, the limit immersion $f$ is unbranched.
• When the Riemann surfaces diverge in moduli space, the liminf of the Willmore energy is bounded below by $\text{min}(8\tilde{ m extbackslash{}pi}, \omega_p^n)$, where $\omega_p^n$ is the infimum of energy over disconnected components with total genus $p$.
• The energy threshold $\omega_p^n$ is optimal and generalizes the codimension-3 result of [K-S2] to arbitrary $n$.
• The paper confirms that $\omega_p^n > 8\tilde{ m extbackslash{}pi}$ for large $p$, consistent with $\beta_p^n \to 8\tilde{ m extbackslash{}pi}$ as $p \to \infty$.
• The analysis shows that isolated singularities of $W^{2,2}$ conformal immersions with finite area and $L^2$ second fundamental form are weak branch points, extending the regularity theory to higher codimensions.