The Liouville property on products of Riemannian manifolds
This short communication is full of content and well worth reading. First, the author gives an ultra-short summary of the major results regarding the validity and the non-validity of the \(c\)-Liouville property on a Riemannian manifold \(M\) where \(c\) denotes the real non-negative potential of the stationary Schrödinger equation \[ \Delta u-c(x)u\;=\;0.\tag{1} \] Recall that the \(c\)-Liouville property means that all bounded solutions of ({1}) vanish identically. Main sources of the summary are the seminal reviews by \textit{E. B. Davies} [Bull. Lond. Math. Soc. 17, 417--436 (1985; Zbl 0583.35032)] and \textit{A. A. Grigor'yan} [J. Sov. Math. 51, No. 3, 2340--2349 (1990); translation from Tr. Semin. Im. I. G. Petrovskogo 14, 66--77 (1989; Zbl 0744.58070); Bull. Am. Math. Soc., New Ser. 36, No. 2, 135--249 (1999; Zbl 0927.58019)]. Basically, everything is well understood in two extreme cases, namely when \(c\) is smooth with compact support -- then the \(c\)-Liouville property holds if and only if \(M\) is of parabolic type; and \(c\) is constant and nowhere vanishing -- then the failure of the \(c\)-Liouville property is equivalent to the stochastic incompleteness of \(M\) (i.e., finite life time of the Wiener process on \(M\)). It is generally believed that a further understanding of the \(c\)-Liouville property may contribute substantially to the classification of non-compact Riemannian manifolds. The author's own contribution in this note is to prove a particular case of a conjecture by A. A. Grogor'yan regarding the preservation of the \(c\)-Liouville property of a non-compact Riemannian manifold under the product with a compact Riemannian manifold regarding a suitably constructed product potential. The proof is based on a previous result of \textit{A. A. Grigor'yan} and \textit{N. S. Nadirashvili} [Sov. Math. 31, No. 5, 31--42 (1987); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1987, No. 5 (300), 25--33 (1987; Zbl 0665.58040)] on variational stability of the Liouville property.
scientific article; zbMATH DE number 6138384