Prescription of finite Dirichlet eigenvalues and area on surface with boundary

TL;DR

This paper demonstrates that for any finite sequence of eigenvalues and a specified area, one can construct a metric on a compact surface with boundary to realize these eigenvalues and area simultaneously.

Contribution

It establishes the existence of metrics on surfaces with boundary that prescribe finite Dirichlet eigenvalues and area, extending spectral geometry results to boundary cases.

Findings

Existence of metrics with prescribed Dirichlet eigenvalues and area.

Construction method for metrics on surfaces with boundary.

Extension of spectral prescription results to bounded surfaces.

Abstract

In the present paper, we consider Dirichlet Laplacian on compact surface. We show that for a fixed surface with boundary $X$, a finite increasing sequence of real numbers $0<a_1<a_2<\cdots<a_N$ and a positive number $A$, there exists a metric $g$ on $X$ such that for any integer $1\leq k\leq N$, we have $\lambda_k^\mathcal{D}(X,g)=a_k$ and $\mathrm{Area}(X,g)=A$.