For complete non-compact surfaces admitting a finite-index Schrödinger operator with non-negative potential, the curvature at infinity of the Fischer-Colbrie metric controls critical points, number of ends, and plane rigidity.
Cohn-Vossen,K¨ urzeste Wege und Totalkr¨ ummung auf Fl¨ achen, Composi- tio Math.2(1935), 69–133
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Curvature at Infinity Governs the Topology of Complete Non-Compact Surfaces Admitting Schr\"odinger Operators of Finite Index
For complete non-compact surfaces admitting a finite-index Schrödinger operator with non-negative potential, the curvature at infinity of the Fischer-Colbrie metric controls critical points, number of ends, and plane rigidity.