[Paper Review] An analogue of minimal surface theory in SL(n,C)/SU(n)

Masatoshi Kokubu, Masaro Takahashi+2 more|ArXiv.org|Aug 2, 2000

Geometric Analysis and Curvature Flows5 references4 citations

TL;DR

This paper develops a non-commutative analogue of minimal surface theory in the symmetric space SL(n,C)/SU(n), generalizing both Euclidean minimal surfaces and constant mean curvature one surfaces in hyperbolic 3-space. It establishes a Weierstrass-type representation formula using null holomorphic maps into SL(n,C) and proves a Chern-Osserman-type inequality for surfaces with holomorphic right Gauss maps, extending classical results to higher-rank symmetric spaces.

ABSTRACT

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for such surfaces are given.

Motivation & Objective

• To generalize minimal surface theory beyond Euclidean space and hyperbolic 3-space to higher-rank symmetric spaces such as SL(n,C)/SU(n).
• To identify a class of surfaces in SL(n,C)/SU(n) that inherit key properties of minimal and CMC-1 surfaces, including conformal Gauss maps and isometric realizations.
• To construct a Weierstrass-type representation formula for these surfaces using holomorphic data and null immersions into SL(n,C).
• To establish a Chern-Osserman-type inequality for the total absolute curvature of such surfaces, extending classical results to non-compact duals of compact semi-simple Lie groups.

Proposed method

• Define the right Gauss map ν_R: M → P(ad(g)) via the projection of f_z f^{-1} for a conformal immersion f: M → G/H, where G = SL(n,C) and H = SU(n).
• Characterize surfaces with holomorphic right Gauss maps as those for which the mean curvature vector length is proportional to the sectional curvature of the ambient space.
• Construct a Weierstrass-type representation using a g-valued holomorphic 1-form α satisfying B(α,α) = 0 and -B(α,σ(α)) > 0, where σ is the involution for the symmetric pair (G,H).
• Obtain the surface by solving F^{-1}dF = α and projecting F: M̃ → G to G/H, where M̃ is the universal cover of M.
• Define the dual surface f^# via the inverse of the null holomorphic map F, which is multi-valued on M but has a single-valued first fundamental form.
• Apply Frobenius series methods to analyze the monodromy and behavior at ends, using the Fuchsian system with regular singularities to derive asymptotic expansions.

Experimental results

Research questions

• RQ1 How can minimal surface theory be generalized to non-compact duals of compact semi-simple Lie groups such as SL(n,C)/SU(n)?
• RQ2 What conditions on a conformal immersion into SL(n,C)/SU(n) ensure that its right Gauss map is holomorphic, and what geometric properties do such surfaces possess?
• RQ3 Can a Weierstrass-type representation formula be constructed for these surfaces using holomorphic data and null immersions into SL(n,C)?
• RQ4 What is the analog of the Chern-Osserman inequality for surfaces with holomorphic right Gauss maps in SL(n,C)/SU(n), and how does it constrain total curvature?

Key findings

• Surfaces in SL(n,C)/SU(n) with holomorphic right Gauss maps generalize both minimal surfaces in R^n and CMC-1 surfaces in H^3, inheriting their key conformal and isometric properties.
• A Weierstrass-type representation formula exists: any such surface arises as the projection of a null holomorphic map F: M̃ → SL(n,C) solving F^{-1}dF = α for a holomorphic 1-form α with B(α,α) = 0 and -B(α,σ(α)) > 0.
• The dual surface f^# is defined via F^{-1} and is multi-valued on M, but its first fundamental form is single-valued, generalizing the duality in CMC-1 surface theory.
• For surfaces with finite total absolute curvature, the ends are isolated and the monodromy is governed by a Fuchsian system with regular singularities, allowing asymptotic analysis via Frobenius series.
• The Chern-Osserman-type inequality holds: the total absolute curvature of f or its dual f^# is bounded below by a topological invariant involving the genus and number of ends.
• When the monodromy matrix R has eigenvalues with non-integer real parts, solutions to the associated Fuchsian system admit Frobenius-type expansions with logarithmic terms, and the number of independent solutions matches the algebraic multiplicity of the eigenvalue λ₀.