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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