In this paper, we study finite projective differential modules on $p$-adic polyannuli satisfying the Robba condition. Christol and Mebkhout proved the decomposition theorem (the $p$-adic Fuchs theorem) of such differential modules on one dimensional $p$-adic annuli under certain non-Liouvilleness assumption and Gachet generalized it to higher dimensional cases. On the other hand, Kedlaya proved a generalization of the $p$-adic Fuchs theorem in one dimensional case. We prove Kedlaya's generalized version of $p$-adic Fuchs theorem in higher dimensional cases.

The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, (Tohoku Math. J. 72 (2020), 349--378), in detail. We prove that the coefficient matrix can be decomposed as a product of diagonal matrices each in a single variable and of orthogonal matrices with constant entries for a general homogeneous cone. Moreover, under a suitable condition, we show that the associated zeta functions admit a kind of completion forms.

Daemi and Scaduto gave an extension of the slice-ribbon conjecture which asks whether a smooth concordance between a knot and a torus knot induces a ribbon concordance between them with an appropriate direction. In this paper, we characterize the two-bridge knots smoothly concordant to non-trivial torus knots via a local move called $t_n$-move. As an application, we confirm that Daemi and Scaduto's extension of the slice-ribbon conjecture is true for all two-bridge knots.

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\mathrm{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a diffeomorphic splitting (or isometric splitting in the particular class of Berwald spaces). This splitting phenomenon is comparable to the Cheeger-Gromoll type splitting theorem by Ohta. We also obtain the rigidity results of logarithmic Sobolev and Bakry--Ledoux isoperimetric inequalities via needle decomposition as corollaries.

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the rigidity models of the Shannon entropy power are Einstein or quasi Einstein manifolds with Hessian solitons. Moreover, we prove the NFW formula which indicates the intrinsic relationship between the Shannon entropy power $\mathcal{N}$, Perelman's $\mathcal{F}$-functional and $\mathcal{W}$-entropy on the Ricci flow. This leads us to prove that the Shannon entropy power for the conjugate heat equation on Ricci flow is convex and the corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with non-negative ($m$-dimensional Bakry-Emery) Ricci curvature and the maximal volume growth condition.

We are interested in establishing global stability results for a system of semilinear wave and Klein-Gordon equations with mixed coupling nonlinearities, that is, we consider all of the possible quadratic nonlinear terms of the type of wave and Klein-Gordon interactions. The main difficulties are due to the absence of derivatives on the wave component in the nonlinearities. By doing a nonlinear transformation on the wave equation, we reveal a hidden null structure. Next, by using the scaling vector field on the wave component only, which is generally avoided, we are able to get good $L^2$-type estimates on the wave component. Then we distinguish high-order and low-order energy of both wave and Klein-Gordon components, which allows us to close the bootstrap argument.