Real forms of complex surfaces of constant mean curvature

Abstract

It is known that complex constant mean curvature ( <sc>CMC</sc> for short) immersions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb C^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are natural complexifications of <sc>CMC</sc> -immersions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, conversely we consider <italic>real form surfaces</italic> of a complex <sc>CMC</sc> -immersion, which are defined from real forms of the twisted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German l left-parenthesis 2 comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {sl}(2, \mathbb C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> loop algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda German s German l left-parenthesis 2 comma double-struck upper C right-parenthesis Subscript sigma"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi> σ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda \mathfrak {sl}(2, \mathbb C)_\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and classify all such surfaces according to the classification of real forms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda German s German l left-parenthesis 2 comma double-struck upper C right-parenthesis Subscript sigma"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi> σ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda \mathfrak {sl}(2, \mathbb C)_\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . There are seven classes of surfaces, which are called <italic>integrable surfaces</italic> , and all integrable surfaces will be characterized by the (Lorentz) harmonicities of their Gauss maps into the symmetric spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S squared"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1 comma 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{1,1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -symmetric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper C right-parenthesis slash upper U left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mm