Minimal surfaces in H ³ (−1) via Willmore geometry, Gauss maps, meromorphic potentials, and algorithmic constructions

This article develops a self-contained account of minimal surfaces in hyperbolic three-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> H³(-1) from the vantage point of Willmore geometry and the DPW (loop-group) method. We begin with the Hermitian model <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≃</m:mo> <m:mi mathvariant="normal">S</m:mi> <m:mi mathvariant="normal">L</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mi mathvariant="normal">S</m:mi> <m:mi mathvariant="normal">U</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> H³(-1) SL(2,C)/SU(2) and the minimal Lax pair, and then place the classical data alongside the Bryant–Epstein hyperbolic Gauss maps, Kokubu’s adjusted normal Gauss map, and Hélein’s roughly harmonic lifts. From this perspective, we formulate a DPW-type representation framework for minimal immersions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> H³(-1) based on meromorphic rank-one λ −1 -potentials, carefully distinguishing the unitary loop factor from the non-unitary frame used to recover the hyperbolic immersion. We explain how monodromy, unitarization, and completeness enter the global picture. A practical algorithm tailored to the minimal case | λ | = 1 is presented together with potential-level examples and a concise numerical appendix (Magnus integrator, polar/Iwasawa factorization, and unitarization strategies).