On manifolds with almost non-negative Ricci curvature and integrally-positive $$k^{th}$$-scalar curvature

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest <i>k</i> eigenvalues of the Ricci tensor. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>M</mml:mi> <mml:mi>n</mml:mi></mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is a Riemannian manifold satisfying such curvature bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , then we show that <i>M</i> is contained in a neighbourhood of controlled width of an isometrically embedded 1-dimensional sub-manifold. From this, we deduce several metric and topological consequences: <i>M</i> has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of <i>M</i> is bounded above by 1, and there is precise information on elements of infinite order in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>π</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> . If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>M</mml:mi> <mml:mi>n</mml:mi></mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is a Riemannian manifold satisfying such bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> , then we show that <i>M</i> has at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -dimensional behavior at large scales. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mtext>dim</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , so that the integral lower bound is on the scalar curvature, assuming in addition that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> . From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.