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- W4288621240 abstract "Let $M$ be a closed smooth manifold. In 1999, L. Friedlander and N. Nadirashvili introduced a new differential invariant $I_1(M)$ using the first normalized nonzero eigenvalue of the Lalpace-Beltrami operator $Delta_g$ of a Riemannian metric $g$. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use $k$-th eigenvalues of $Delta_g$ to define the invariants $I_k(M)$ indexed by positive integers $k$. In the present paper the values of these invariants on surfaces are investigated. We show that $I_k(M)=I_k(mathbb{S}^2)$ unless $M$ is a non-orientable surface of even genus. For orientable surfaces and $k=1$ this was earlier shown by R. Petrides. In fact L. Friedlander and N. Nadirashvili suggested that $I_1(M)=I_1(mathbb{S}^2)$ for any surface $M$ different from $mathbb{RP}^2$. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has $I_k(M)>I_k(mathbb{S}^2)$. We also discuss the connection between the Friedlander-Nadirashvili invariants and the theory of cobordisms, and conjecture that $I_k(M)$ is a cobordism invariant." @default.
- W4288621240 created "2022-07-30" @default.
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- W4288621240 date "2019-01-27" @default.
- W4288621240 modified "2023-10-17" @default.
- W4288621240 title "On the Friedlander-Nadirashvili invariants of surfaces" @default.
- W4288621240 doi "https://doi.org/10.48550/arxiv.1901.09443" @default.
- W4288621240 hasPublicationYear "2019" @default.
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