Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum

TL;DR

This paper demonstrates that on any compact manifold with boundary, one can construct Riemannian metrics with a prescribed finite set of Dirichlet eigenvalues and a specified volume, showcasing control over spectral properties.

Contribution

It establishes the existence of Riemannian metrics with prescribed finite Dirichlet spectrum and volume on arbitrary compact manifolds with boundary.

Findings

Existence of metrics with prescribed eigenvalues and volume

Control over spectral properties of manifolds

Applicable to manifolds of dimension n ≥ 3

Abstract

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real numbers $0<a_1<a_2 \leq a_3 \leq \cdots \leq a_N$ and any positive number $V$, there exists a Riemannian metric $g$ on $M$ such that $\mathrm{Vol}(M,g)=V$ and $\lambda^\mathcal{D}_k(M,g)=a_k$ for any integer $1 \leq k \leq N$.