An analogue of minimal surface theory in \(\operatorname{SL}(n,\mathbb{C})/\operatorname{SU}(n)\) (Q2781362)

The authors study so-called Weierstrass-Bryant type representation formulas that generalize the classical Weierstrass formula for minimal surfaces in \({\mathbb R}^3\) as well as \textit{R.~Bryant}'s representation formula for surfaces of constant mean curvature \(h=1\) in the hyperbolic space \(H^3\), which will be named CMC-1-surfaces for short [Astérisque 154/155, 321-347 (1988; Zbl 0635.53047)], to immersions \(f\) of a Riemann surface \(M\) into the homogeneous space \(\text{SL}(n,{\mathbb C})/\text{SU}(n)\). The existence of such generalizations can be imagined when taking in consideration that \(H^3\) can be identified with \(\text{SL}(2,{\mathbb C})/\text{SU}(2)\). Both, the Weierstrass- and the Bryant-formula assign to pairs \((g,\omega)\) consisting of a meromorphic function \(g\) and a meromorphic 1-form \(\omega\) on \(M\) minimal (resp. CMC-1) immersions \(f\) of \(M\) into \({\mathbb R}^3\) (resp. \(H^3\)). Such an \(f\) is defined via an intermediate map \(F\) of \(M\) into \({\mathbb C}^3\) (resp. \(\text{SL}(2,{\mathbb C})\)) given by NEWLINE\[NEWLINEdF={1\over 2} \begin{pmatrix} 1-g^2\\ i(1+g^2)\\ 2g\end{pmatrix} \omega\text{ resp. } dF=F X,\text{ where } X=\begin{pmatrix} g &-g^2\\ 1 &-g\end{pmatrix}\omega. NEWLINE\]NEWLINE Then, \(f\) is obtained as \(f=\Re\circ F\) in the first and as \(f=F F^\ast\) in the second case, where \(F^\ast=\bar{F}^\top\) is the adjoint matrix of \(F\). In both cases the Gauss map (resp. the hyperbolic Gauss map) of the immersion \(f\) is a conformal mapping from \(M\) to the two-dimensional sphere \(S^2\) (resp. \(S^2_\infty\)) that coincides with \(g\) if \(S^2\) and \(S^2_\infty\) are parametrized in a suitable way, for instance by stereographic projection in the case of \(S^2\). Moreover, minimal surfaces and CMC-1-surfaces that correspond to the same data \((g,\omega)\) are locally isometric, i.e., the Riemannian metrics induced by the two corresponding immersions \(f\) agree. NEWLINENEWLINENEWLINELet \(G\) be a complex semisimple Lie group, \(\mathfrak{g}\) the Lie algebra, and \(H\) a compact real form of \(G\). The quotient space \(G/H\) has the structure of a Riemannian symmetric space. By the Cartan embedding theorem one can assume that \(G/H\subset G\) and \(G\) can be regarded as embedded in \(\text{SL}(n,{\mathbb C})\) by the adjoint representation. The right Maurer-Cartan derivative \(f_z f^{-1}\) of an immersion \(f:M\rightarrow G/H\subset G\subset\text{SL}(n,{\mathbb C})\) is regarded as a map \(f_z f^{-1}:M\rightarrow\mathfrak{gl}(n,{\mathbb C})\) into the Lie algebra of \(\text{SL}(n,{\mathbb C})\). Let \(\pi:\mathfrak{gl}(n,{\mathbb C})\rightarrow P(\mathfrak{gl}(n,{\mathbb C}))\) be the natural projection onto the projective space. The right Gauss map of \(f\) is defined as the composition \([f_z f^{-1}]=\pi\circ f_z f^{-1}\). NEWLINENEWLINENEWLINEA fundamental result is the following: NEWLINENEWLINENEWLINETheorem 1: Denote by \(B\) the Killing form of \(\mathfrak{g}\) and by \(\sigma:\mathfrak{g}\rightarrow\mathfrak{g}\) the involution corresponding to the symmetric pair \((G,H)\). Let \(\alpha\) be a \(\mathfrak{g}\)-valued holomorphic 1-form on \(M\) such that \(B(\alpha,\alpha)=0\) and \(-B(\alpha,\sigma(\alpha))>0\). If \(F:M\rightarrow G\) is a solution of the differential equation \(F^{-1} dF=\alpha\), then \(F\) followed by the projection of \(G\) onto \(G/H\) is a conformal immersion into \(G/H\) whose right-Gauss-map is holomorphic. Conversely, any conformal immersion \(f\) into \(G/H\) with holomorphic right-Gauss-map can be defined by such an \(\alpha\). NEWLINENEWLINENEWLINEAs an application of the representation formula the authors prove the non-existence of compact surfaces without boundary in \(G/H\) whose Gauss map is holomorphic. The dual immersion \(f^\sharp\) of \(f\) is defined as the immersion corresponding to \(F^{-1}\) and its properties are discussed. One obtains the following Chern-Ossermann type inequality: NEWLINENEWLINENEWLINETheorem 2: Let \(f:M\rightarrow G/H\) be a complete conformal immersion with holomorphic right Gauss map, \(f^\sharp\) its dual immersion, \(K^\sharp\) and \(dA^\sharp\) the corresponding Gauss curvature and surface area element respectively. Then NEWLINE\[NEWLINE{1\over 2\pi}\int_M-K^\sharp dA^\sharp\geq-\chi(M)+ \text{ number of ends of }M.NEWLINE\]