arXiv subjects

Yuxiu Lu

Publications and source records attributed to Yuxiu Lu.

5 recordsLinked to original sources

arXiv2026

An analytic approach to $P$-adic diffeomorphism group and Teichm\"{u}ller theory

We consider a specific class of infinite dimensional $p$-adic Lie groups, i.e., a sort of diffeomorphism groups on $p$-adic ball $\operatorname{Diff}^{\operatorname{an}}(B_\epsilon)$. It turns out that this group has a natural logarithmic structure that leads to a $p$-adic version of Teichm\"{u}ller theory on diffeomorphism groups, which also presents some remarkable hydrodynamic facets. We further apply this framework to Mochizuki's $p$-adic Teichm\"{u}ller theory and Inter-universal Teichm\"{u}ller theory (IUT), and give a new reformulation of IUT as a Teichm\"{u}ller theory on automorphisms of two-dimensional group schemes.

math.NT↗

arXiv2025

The $L^p$-geometry and its applications

We generalize the classical Fisher information metric on statistical models to $L^p$-metrics on various spaces of differential forms or group of diffeomorphisms. Using this new interpretation from information geometry, we derive several new results in geometry on group of diffeomorphisms, symplectic geometry and Teichm\"{u}ller theory. This includes geometry of $\operatorname{Diff}_{-\infty}(\RR)$, similar to that of universal Teichm\"{u}ller space in essence, also a study on the space of all symplectic forms on a symplectic manifold $M$ and a generalization of Gelfand-Fuchs cocycles to higher-dimension. Furthermore, we answer questions in $\alpha$-geometry posed by Gibilisco, and generalize the $L^p$-metrics to geometry on Orlicz spaces.

math.AT↗

arXiv2023

The $L^p$-Fisher-Rao metric and Amari-Cencov $α$-connections

We introduce a family of Finsler metrics, called the $L^p$-Fisher-Rao metrics $F_p$, for $p\in (1,\infty)$, which generalizes the classical Fisher-Rao metric $F_2$, both on the space of densities Dens$_+(M)$ and probability densities Prob$(M)$. We then study their relations to the Amari-uCencov $α$-connections $\nabla^{(α)}$ from information geometry: on Dens$_+(M)$, the geodesic equations of $F_p$ and $\nabla^{(α)}$ coincide, for $p = 2/(1-α)$. Both are pullbacks of canonical constructions on $L^p(M)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of $α$-geodesics as being energy minimizing curves. On Prob$(M)$, the $F_p$ and $\nabla^{(α)}$ geodesics can still be thought as pullbacks of natural operations on the unit sphere in $L^p(M)$, but in this case they no longer coincide unless $p=2$. Using this transformation, we solve the geodesic equation of the $α$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of $F_p$, and study their relation to $\nabla^{(α)}$.

math.DG↗

arXiv2023

A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations

We show that two families of equations on the real line, the generalized inviscid Proudman--Johnson equation, and the $r$-Hunter--Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman--Johnson equations as geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on an appropriate diffeomorphism group on $\mathbb{R}$. Generalizing a construction of Lenells for the Hunter--Saxton equation, we analyze the $r$-Hunter--Saxton equation using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equation on the $L^r$-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.

math.DG↗

arXiv2017

On cobordism of generalized (real) Bott manifolds

We show that all generalized (real) Bott manifolds which are (small covers) quasitoric manifolds over a product of simplices $Δ^{n_1}\times\cdots\timesΔ^{n_r}\timesΔ^{1}$ are always boundaries of some manifolds. But these manifolds with the natural $(\mathbb{Z}_2)^n$ action do not necessarily bound equvariantly. In addition, we can construct some examples of null-cobordant but not orientedly null-cobordant manifolds among quasitoric manifolds.

math.AT↗