arXiv subjects

Yasuaki Fujitani

Publications and source records attributed to Yasuaki Fujitani.

9 recordsLinked to original sources

arXiv2026

Comparison theorems in centro-affine differential geometry

For weighted manifolds with Bakry-Émery Ricci curvature bounded from below, comparison geometric results have been established. The Bakry-Émery Ricci curvature depends on the effective dimension $N$. In this paper, the case $N = 1$ is further studied in the setting of centro-affine differential geometry. Several examples, such as rotationally symmetric ellipsoids and hyperboloids, arise in rigidity phenomena of comparison theorems.

math.DG↗

arXiv2026

Stability of weighted minimal hypersurfaces under a lower $1$-weighted Ricci curvature bound

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci curvature bound. We prove a Schoen-Yau type criterion, and conclude a structure theorem for three-dimensional weighted manifolds of non-negative $1$-weighted Ricci curvature. We also show non-existence results under volume growth conditions, and conclude smooth compactness theorems.

math.DG↗

arXiv2026

Choi-Wang inequality for affine connections

Choi-Wang established a lower bound for the first non-zero eigenvalue of the Laplacian on minimal hypersurfaces in manifolds with positive Ricci curvature. We extend this Choi-Wang type inequality to the setting of positive Ricci curvature with respect to the Li-Xia type affine connection.

math.DG↗

arXiv2025

Geometric analysis on weighted manifolds under lower $0$-weighted Ricci curvature bounds

We develop geometric analysis on weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi-Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.

math.DG↗

arXiv2025

A Steklov eigenvalue estimate for affine connections and its application to substatic triples

Choi-Wang obtained a lower bound of the first eigenvalue of the Laplacian on closed minimal hypersurfaces. On minimal hypersurfaces with boundary, Fraser-Li established an inequality giving a lower bound of the first Steklov eigenvalue as a counterpart of the Choi-Wang type inequality. These inequalities were shown under lower bounds of the Ricci curvature. In this paper, under non-negative Ricci curvature associated with an affine connection introduced by Wylie-Yeroshkin, we give a generalization of Fraser-Li type inequality. Our results hold not only for weighted manifolds under non-negative $1$-weighted Ricci curvature but also for substatic triples.

math.DG↗

arXiv2023

Analysis of harmonic functions under lower bounds of $N$-weighted Ricci curvature with $\varepsilon$-range

The behavior of harmonic functions on Riemannian manifolds under lower bounds of the Ricci curvature has been studied from both analytic and geometric viewpoints. For example, some Liouville type theorems are obtained under lower bounds of the Ricci curvature. Recently, those results are generalized under lower bounds of the $N$-weighted Ricci curvature $\mbox{Ric}_ψ^N$ with $N \in [n,\infty]$. In this paper, we present a Liouville type theorem for harmonic functions of sublinear growth and a gradient estimate of harmonic functions on weighted Riemannian manifolds under weaker lower bounds of $\mbox{Ric}_ψ^N$ with $N < 0$. We also prove an $L^p$-Liouville theorem under lower bounds of $\mbox{Ric}_ψ^N$ with $N \in [n,\infty]$ in a way different from that of [Wu, 2014]. Our results are obtained as a consequence of an argument under lower bounds of $\mbox{Ric}_ψ^N$ with $\varepsilon$-range, which is a unification of constant and variable curvature bounds. Among various methods considered for the analysis of harmonic functions, this paper focuses on methods using the Moser's iteration procedure.

math.DG↗

arXiv2023

Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of $N$-weighted Ricci curvature with $N < 0$

The Aronson-Bénilan gradient estimate for the porous medium equation has been studied as a counterpart to the Li-Yau gradient estimate for the heat equation. In this paper, we give the Aronson-Bénilan gradient estimates for the porous medium equation on weighted Riemannian manifolds under lower bounds of $N$-weighted Ricci curvature with $\varepsilon$-range for some $N < 0$. This is a generalization of those estimates under constant lower $N$-weighted Ricci curvature bounds with $N\in [n,\infty)$.

math.DG↗

arXiv2022

Some functional inequalities under lower Bakry-Émery-Ricci curvature bounds with $\varepsilon$-range

For $n$-dimensional weighted Riemannian manifolds, lower $m$-Bakry-Émery-Ricci curvature bounds with $\varepsilon$-range, introduced by Lu-Minguzzi-Ohta, integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower $m$-Bakry-Émery-Ricci curvature bounds with $\varepsilon$-range. These generalize those inequalities under constant curvature bounds for $m \in (n,\infty)$ to $m\in(-\infty,1]\cup\{\infty\}$.

math.DG↗

arXiv2022

Information geometry of warped product spaces

Information geometry is an important tool to study statistical models. There are some important examples in statistical models which are regarded as warped products. In this paper, we study information geometry of warped products. We consider the case where the warped product and its fiber space are equipped with dually flat connections and, in the particular case of a cone, characterize the connections on the base space $\mathbb{R}_{>0}$. The resulting connections turn out to be the $α$-connections with $α= \pm{1}$.

math.DG↗