arXiv subjects

Masayuki Aino

Publications and source records attributed to Masayuki Aino.

5 recordsLinked to original sources

arXiv2021

Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the $ε$-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is $O\left(\left(\log n/n\right)^{1/(m+2)}\right)$, where $m$ and $n$ denote the dimension of the manifold and the sample size, respectively.

math.DG↗

arXiv2021

Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

math.DG↗

arXiv2021

Lichnerowicz-Obata Estimate, Almost Parallel $p$-form and Almost Product Manifolds

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.

math.DG↗

arXiv2019

Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.

math.DG↗

arXiv2017

Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants $\{Ω_k(g)\}$ for closed Riemannian manifolds. The value of $Ω_k(g)$ delicately reflects the spherical part of the manifold. Indeed, $Ω_1(g)$ and $Ω_2(g)$ characterize the standard sphere.

math.DG↗