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We consider the principal eigenvalue problem for the Laplace-Beltramioperator on the upper half of a topological torus under the Dirichlet boundarycondition. We present a construction of the upper half of a topological torusthat admits the principal eigenfunction having exact numbers of criticalpoints. Furthermore, we manage to identify the locations of all the criticalpoints of the principal eigenfunction explicitly.

The principal eigenfunction of the Dirichlet Laplacian with prescribed numbers of critical points on the upper half of a topological torus