We prove several analogs of Gromov’s macroscopic dimension conjecture with extra curvature assumptions. More explicitly, we show that for an open Riemannian n-manifold (M, g) of nonnegative Ricci (resp. sectional) curvature, if it has uniformly positive scalar curvature and it is uniformly volume noncollapsed, then the essential (resp. Hausdorff) dimension of an asymptotic cone, as a notion of largeness, has a sharp upper bound <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3139_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which is 2 less than the upper bound for an open Riemannian manifold with only nonnegative Ricci curvature. As a consequence, the dimension of space of linear growth harmonic functions of M has upper bound <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3139_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> which is also 2 less than the sharp bound <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3139_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> when M only has nonnegative Ricci curvature. We also prove the first Betti number upper bound is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3139_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> if M is compact, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3139_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$n-3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> if M is non-compact. When M is compact we show a fibration theorem over torus, and a rigidity theorem for the fiber when the first Betti number upper bound is achieved.

Two-dimension vanishing, splitting and positive scalar curvature

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2025年4月25日

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Xingyu Zhu

摘要

We prove several analogs of Gromov’s macroscopic dimension conjecture with extra curvature assumptions. More explicitly, we show that for an open Riemannian n-manifold (M, g) of nonnegative Ricci (resp. sectional) curvature, if it has uniformly positive scalar curvature and it is uniformly volume noncollapsed, then the essential (resp. Hausdorff) dimension of an asymptotic cone, as a notion of largeness, has a sharp upper bound $n-2$ $n - 2$ , which is 2 less than the upper bound for an open Riemannian manifold with only nonnegative Ricci curvature. As a consequence, the dimension of space of linear growth harmonic functions of M has upper bound $n-1$ $n - 1$ which is also 2 less than the sharp bound $n+1$ $n + 1$ when M only has nonnegative Ricci curvature. We also prove the first Betti number upper bound is $n-2$ $n - 2$ if M is compact, and $n-3$ $n - 3$ if M is non-compact. When M is compact we show a fibration theorem over torus, and a rigidity theorem for the fiber when the first Betti number upper bound is achieved.