arXiv subjects

Kazumasa Narita

Publications and source records attributed to Kazumasa Narita.

4 recordsLinked to original sources

arXiv2026

Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds

For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $λ_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant quantity $λ_{k}(g)\mbox{Vol}(M,g)^{2/n}$ as a functional over all the metrics in a fixed conformal class, we derive a second variation formula for the functional. As a corollary, we prove that if the canonical flat metric on a torus is such that the multiplicity of $λ_{1}$ is two, then the flat metric is not a maximal point of the functional in its conformal class. This is a higher dimensional extension of Karpukhin's very recent work.

math.DG↗

arXiv2025

Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics $(g_{t})_{t > 0}$ on $M$, which is called the canonical variation. Let $λ_{1}(g_{t})$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ^{M}_{g_{t}}$ and $\mbox{Vol}(M,g_{t})$ the volume of $(M, g_{t})$. In 1982, Bérard-Bergery and Bourguignon showed that the scale-invariant quantity $λ_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $0$ with $t$. In this paper, we show that if each fiber is Einstein and $(M,g)$ satisfies a certain condition about its Ricci curvature, then bounds for $λ_{1}(g_{t})$ can be obtained. In particular this implies $λ_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $\infty$ with $t$. Moreover, using the bounds, we consider stability of critical points of the Yamabe functional. We will see that our results can be applied to many examples. In particular, we consider the twistor fibration of a quaternionic Kähler manifold of positive scalar curvature.

math.DG↗

arXiv2024

Deformation of Kähler Metrics and an Eigenvalue Problem for the Laplacian on a Compact Kähler Manifold

We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the $k$-th eigenvalue $λ_{k}$ as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of $λ_{k}$-extremal Kähler metric. We deduce a condition for a Kähler metric to be $λ_{k}$-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.

math.DG↗

arXiv2024

Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices

It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with eigenvalues between $0$ and $1$ and summing to $k$. We give a new proof of this fact. We also give an existence theorem for a certain combinatorial class of hypermatrices by a similar argument. This existence theorem can be rewritten into an existence theorem for a uniform weighted hypergraph with given weighted degree sequence.

math.CO↗