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Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum

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Abstract

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

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Xiang He & Zuoqin Wang

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  1. Xiang He
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  2. Zuoqin Wang
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Correspondence to Zuoqin Wang.

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Communicated by F.-H. Lin.

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Partially supported by National Key R and D Program of China 2020YFA0713100, and by NSFC No. 12171446, 11721101.

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He, X., Wang, Z. Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum. Calc. Var. 63, 78 (2024). https://doi.org/10.1007/s00526-024-02684-x

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