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<p>We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold <i>M</i>. We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {C}^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>–Riemannian metrics on <i>M</i> or the set of all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {C}^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>–diffeomorphisms on a domain into <i>M</i>. In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase.</p>

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Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月16日

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  • C. L. Cunha,
  • J. N. V. Gomes,
  • M. A. M. Marrocos

摘要

We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold M. We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all \(\mathcal {C}^r\) C r –Riemannian metrics on M or the set of all \(\mathcal {C}^r\) C r –diffeomorphisms on a domain into M. In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase.