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<p>Choi and Wang provided an estimate for the lower bound of the first eigenvalue for closed minimal surfaces in a complete three-dimensional Riemannian manifold with positive Ricci curvature. Following their work, Cheng–Mejia–Zhou and Ding–Xin generalized these estimates to closed <i>f</i>-minimal surfaces and closed self-shrinkers, respectively. Beyond closed cases, Brendle–Tsiamis and this paper independently addressed complete non-compact cases. In this paper, we estimate the lower bound of the first eigenvalue for complete non-compact <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-hypersurfaces in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {R}^{n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with the restriction <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\vert \lambda \vert &lt;\frac{1}{2}-\frac{1}{2n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>λ</mi> <mo stretchy="false">|</mo> <mo>&lt;</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Specifically, the first eigenvalue for such <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-hypersurfaces is at least <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\frac{1}{4}-\frac{\lambda ^2}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>-</mo> <mfrac> <msup> <mi>λ</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. This result generalizes Brendle and Tsiamis’ result for complete non-compact self-shrinkers with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\lambda =0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where the first eigenvalue is bounded below by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\frac{1}{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></EquationSource> </InlineEquation>. Considering self-shrinkers as minimal in a smooth metric measure space with a particular weight function, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-hypersurfaces can be viewed as constant mean curvature hypersurfaces in the same space. Consequently, the result in this paper serves as a natural generalization from minimal to constant mean curvature hypersurfaces.</p>

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The First Eigenvalue Estimate for \(\lambda \)-hypersurfaces in Euclidean Space

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Jihyeon Lee

摘要

Choi and Wang provided an estimate for the lower bound of the first eigenvalue for closed minimal surfaces in a complete three-dimensional Riemannian manifold with positive Ricci curvature. Following their work, Cheng–Mejia–Zhou and Ding–Xin generalized these estimates to closed f-minimal surfaces and closed self-shrinkers, respectively. Beyond closed cases, Brendle–Tsiamis and this paper independently addressed complete non-compact cases. In this paper, we estimate the lower bound of the first eigenvalue for complete non-compact \(\lambda \) λ -hypersurfaces in \(\mathbb {R}^{n+1}\) R n + 1 with the restriction \(\vert \lambda \vert <\frac{1}{2}-\frac{1}{2n}\) | λ | < 1 2 - 1 2 n . Specifically, the first eigenvalue for such \(\lambda \) λ -hypersurfaces is at least \(\frac{1}{4}-\frac{\lambda ^2}{2}\) 1 4 - λ 2 2 . This result generalizes Brendle and Tsiamis’ result for complete non-compact self-shrinkers with \(\lambda =0\) λ = 0 , where the first eigenvalue is bounded below by \(\frac{1}{4}\) 1 4 . Considering self-shrinkers as minimal in a smooth metric measure space with a particular weight function, \(\lambda \) λ -hypersurfaces can be viewed as constant mean curvature hypersurfaces in the same space. Consequently, the result in this paper serves as a natural generalization from minimal to constant mean curvature hypersurfaces.