Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A smooth endomorphism field $L:TM\to TM$ is Nijenhuis if, for all smooth vector fields $X,Y$, its torsion \[N_L(X,Y)=L^2[X,Y]+[LX,LY]-L[LX,Y]-L[X,LY]\] vanishes identically.

The Segre characteristic of a real endomorphism records its Jordan block sizes, the grouping of blocks by equal eigenvalues, and which eigenvalues are real or occur in complex-conjugate pairs. Numerical eigenvalues may vary while this characteristic stays fixed. An endomorphism field on $\mathbb R^n$ has compact support if it vanishes outside a compact subset.

Statement recorded in the supplied source.

For an arbitrary prescribed Segre characteristic in dimension $n$, does there exist a smooth compactly supported Nijenhuis field $L$ on $\mathbb R^n$ and a nonempty open domain $U\subset\mathbb R^n$ such that $L_x$ has that characteristic for every $x\in U$? More generally, find necessary and sufficient conditions on the characteristic for such an $L$ and $U$ to exist. [S2]

Short English statement

Which fixed real Jordan types can occur on an open region of a compactly supported Nijenhuis tensor? Give a necessary and sufficient criterion.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100509 (AMR-020-0509). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.11 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.