The index growth and multiplicity of closed geodesics

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in \cite{LoD1} on rational and in \cite{DuL1}, \cite{Rad4} and \cite{Rad5} on completely non-degenerate closed geodesics on spheres and \CP^2 to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.Comment: 59 pages, 1 figure, to appear in Journal of Functional Analysis (JFA) and this is the final versio

Three closed characteristics on non-degenerate star-shaped hypersurfaces in R6\mathbf{R}^{6}

In this paper, we prove that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{6}$ which carries no prime closed characteristic of Maslov-type index $0$ or no prime closed characteristic of Maslov-type index $-1$, there exist at least three prime closed characteristics on $\Sigma$.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2205.07082, arXiv:1510.08648, arXiv:2205.1478

Electron correlations and superconductivity in La3_3Ni2_2O7_7 under pressure tuning

Motivated by the recent discovery of superconductivity in La$_3$Ni$_2$O$_7$ under pressure, we discuss the basic ingredients of a model that captures its microscopic physics under pressure tuning. We anchor our description in terms of the spectroscopic evidence of strong correlations in this system. In a bilayer Hubbard model including the Ni $3d$ $x^2-y^2$ and $z^2$ orbitals, we show the ground state of the model crosses over from a low-spin $S=1/2$ state to a high-spin $S=3/2$ state. In the high-spin state, the two $x^2-y^2$ and the bonding $z^2$ orbitals are all close to half-filling, which promotes a strong orbital selectivity in a broad crossover regime of the phase diagram pertinent to the system. Based on these results, we construct an effective multiorbital $t$-$J$ model to describe the superconductivity of the system, and find the leading pairing channel to be an intraorbital spin singlet with a competition between the extended $s$-wave and $d_{x^2-y^2}$ symmetries. Our results highlight the role of strong multiorbital correlation effects in driving the superconductivity of La$_3$Ni$_2$O$_7$.Comment: 5.5+4 pages, 4+3 figures, 2 tables, updated version with supplemental materia