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<p>In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has <i>k</i> ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1919_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </InlineMediaObject> <EquationSource Format="TEX">\(\limsup _{r\rightarrow \infty }\frac{\textrm{Vol}(B(p, r))}{r}\le 4k\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim sup</mo> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <mfrac> <mrow> <mtext>Vol</mtext> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </mfrac> <mo>≤</mo> <mn>4</mn> <mi>k</mi> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound.</p>

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Optimal Asymptotic Volume Ratio for Noncompact 3-Manifolds with Asymptotically Nonnegative Ricci Curvature and a Uniformly Positive Scalar Curvature Lower Bound

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Xian-Tao Huang,
  • Shuai Liu

摘要

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has k ends and finite first Betti number, then it has at most linear volume growth, and furthermore, if the negative part of Ricci curvature decays sufficiently fast at infinity, then we have an optimal asymptotic volume ratio \(\limsup _{r\rightarrow \infty }\frac{\textrm{Vol}(B(p, r))}{r}\le 4k\pi \) lim sup r → ∞ Vol ( B ( p , r ) ) r ≤ 4 k π . In particular, our results apply to 3-dimensional complete non-compact Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound.