[Paper Review] Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

Robert Kusner|ArXiv.org|Jul 19, 2002

Geometric Analysis and Curvature Flows22 references3 citations

TL;DR

This paper establishes that the parabolic classifying map Φ, which assigns to each properly embedded constant mean curvature (CMC) surface its conformal structure and asymptotic necksizes, is a proper, real analytic map from the moduli space of CMC surfaces to a space of parabolic structures. The key result is that Φ has a well-defined (mod 2) degree for genus zero surfaces, which vanishes, implying that each conformal type and necksize configuration is realized by an even number of CMC surfaces—confirming a topological obstruction in the case of triunduloids.

ABSTRACT

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} imes R_+^k$ be the space of parabolic structures over Riemann surfaces of genus g with k (marked) punctures, the real analytic structure coming from the 3g-3+k local complex analytic coordinates on the Riemann moduli space r_{g,k}. Then the parabolic classifying map, Phi: M --> P, which assigns to a CMC surface its induced conformal structure and asymptotic necksizes, is a proper, real analytic map. It follows that Phi is closed and in particular has closed image. For genus g=0, this can be used to show that every conformal type of multiply punctured Riemann sphere occurs as a CMC surface, and -- under a nondegeneracy hypothesis -- that Phi has a well defined (mod 2) degree. This degree vanishes, so generically an even number of CMC surfaces realize any given conformal structure and asymptotic necksizes.

Motivation & Objective

• To understand the global structure of the moduli space of properly embedded constant mean curvature (CMC) surfaces with finite genus and ends.
• To establish a real analytic classifying map Φ from the moduli space of CMC surfaces to the space of parabolic structures on punctured Riemann surfaces.
• To determine the topological and geometric constraints on the realization of conformal types and necksizes by CMC surfaces, particularly in genus zero.
• To investigate the degree of the classifying map Φ and its implications for the multiplicity of CMC surfaces realizing a given conformal and necksize data.

Proposed method

• Define the moduli space M = Mg,k of properly embedded CMC surfaces of genus g with k labeled ends, modulo rigid motions.
• Introduce the space P = Pg,k = Rg,k × Rk+ of parabolic structures on genus g Riemann surfaces with k marked punctures, equipped with a real analytic structure from the 3g−3+k local coordinates on the moduli space Rg,k.
• Construct the parabolic classifying map Φ: M → P that assigns to each CMC surface its induced conformal structure and asymptotic necksizes.
• Prove that Φ is a proper, real analytic map, hence closed with closed image, using transcendental methods and compactness results from prior work.
• Analyze the case g = 0 using the fact that the conformal type of the punctured sphere is unique, and apply the nondegeneracy hypothesis to define the (mod 2) degree of Φ.
• Use the vanishing of the (mod 2) degree to deduce that each conformal type and necksize configuration is realized by an even number of CMC surfaces, generalizing results for triunduloids.

Experimental results

Research questions

• RQ1 What is the image of the parabolic classifying map Φ in the space of parabolic structures, and how do allowable necksizes depend on the underlying conformal type?
• RQ2 Is the preimage Φ−1([Σ]) finite for a given conformal type [Σ], and is 2|χ(Σ)| an upper bound on its cardinality?
• RQ3 Is the moduli space M of CMC surfaces connected, and how does the fundamental group homomorphism induced by Φ relate to the topology of M?
• RQ4 Does M carry a complete metric of nonpositive curvature, making it a K(π,1)-space, and if so, can it be constructed from CMC geometry?
• RQ5 Can one detect whether an immersed surface in R³ is immersed in the sense of Alexandrov using a general obstruction theory?

Key findings

• The parabolic classifying map Φ: M → P is a proper, real analytic map, ensuring that its image is closed.
• For genus g = 0, every conformal type of the multiply punctured Riemann sphere arises as a CMC surface, confirming the existence of such surfaces for all k ≥ 3.
• Under a nondegeneracy hypothesis, the map Φ has a well-defined (mod 2) degree, which vanishes for genus zero surfaces.
• The vanishing of the (mod 2) degree implies that each conformal structure and necksize configuration is realized by an even number of CMC surfaces, generalizing the two-to-one result for triunduloids (k = 3).
• For g = 0 and k = 3, the image of Φ is the simplex defined by the spherical triangle inequalities on necksizes, as previously established in [8].
• The cardinality of Φ−1([Σ]) is bounded above by 2|χ(Σ)| = 2k−2, and this bound is sharp for k = 3, where Φ is two-to-one except at the maximal necksize fold.