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Schrödinger operators associated to a holomorphic map. (English) Zbl 0744.58007
Global differential geometry and global analysis, Proc. Conf., Berlin/Ger. 1990, Lect. Notes Math. 1481, 147-174 (1991).
[For the entire collection see Zbl 0733.00014.]
Let $$\Sigma$$ be a compact Riemann surface, $$S^ 2$$ the unit two-sphere and $$\varphi:\Sigma\to S^ 2$$ a holomorphic mapping. If $$\Sigma$$ is endowed with a metric compatible with the complex structure, one can construct the Laplacian $$\Delta$$, the gradient $$\nabla$$ (depending on the metric of course) and the Schrödinger operator $$L:=\Delta + | \Delta \varphi|^ 2$$.
The authors study those spectral properties of these operators which are related to the map $$\varphi$$ and the surface $$\Sigma$$ and obtain a series of remarkable results mainly on the index of nullity of such holomorphic maps. They also sketch three important related geometric topics where the invariants described appear: the theory of complete minimal surfaces in $$\mathbb{R}^ 3$$, Willmore surfaces, the determinant of the Laplacian of the metrics on a compact surface. Finally, the authors formulate a bunch of open problems.

MSC:
 58C40 Spectral theory; eigenvalue problems on manifolds 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 30F10 Compact Riemann surfaces and uniformization 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature