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Nonexistence of the metric with positive intermediate curvatures on manifolds with boundary

Jingche Chen, Han Hong

TL;DR

This work develops curvature obstruction techniques for compact manifolds with boundary, proving that in dimensions $2\le n\le 7$ and for $1\le m\le n-1$, no metric with positive $m$-intermediate curvature $C_m>0$ exists when the boundary is $m$-convex and a Lefschetz dual cup product of $m$ first cohomology classes remains nonzero on the boundary. The authors introduce stable free boundary weighted $m$-slicings and free boundary $\mu$-bubbles to derive integral obstructions, and they establish rigidity results showing isometric splitting under nonnegativity assumptions. A key extension demonstrates that the obstruction persists under interior or boundary connected sums with arbitrary manifolds, proved via infinite cyclic covers and weighted slicings on $\hat{Y}$. The results generalize prior Geroch-type obstructions to manifolds with boundary and provide both nonexistence and rigidity phenomena in low dimensions, with implications for the geometry and topology of bounded spaces under curvature constraints.

Abstract

We establish curvature obstruction theorems for manifolds with boundary. Our main theorems show that, for dimensions up to 7, a topologically nontrivial compact manifold with boundary cannot have a metric of positive $m$-intermediate curvature if the boundary is $m$-convex, and some rigidity result holds if $m$-intermediate curvature is nonnegative. This non-existence persists after performing a connected sum with an arbitrary manifold. These results generalize results of \cite{brendle,chenshuli,ChuKwongLee,Xu} to manifold with boundary.

Nonexistence of the metric with positive intermediate curvatures on manifolds with boundary

TL;DR

This work develops curvature obstruction techniques for compact manifolds with boundary, proving that in dimensions and for , no metric with positive -intermediate curvature exists when the boundary is -convex and a Lefschetz dual cup product of first cohomology classes remains nonzero on the boundary. The authors introduce stable free boundary weighted -slicings and free boundary -bubbles to derive integral obstructions, and they establish rigidity results showing isometric splitting under nonnegativity assumptions. A key extension demonstrates that the obstruction persists under interior or boundary connected sums with arbitrary manifolds, proved via infinite cyclic covers and weighted slicings on . The results generalize prior Geroch-type obstructions to manifolds with boundary and provide both nonexistence and rigidity phenomena in low dimensions, with implications for the geometry and topology of bounded spaces under curvature constraints.

Abstract

We establish curvature obstruction theorems for manifolds with boundary. Our main theorems show that, for dimensions up to 7, a topologically nontrivial compact manifold with boundary cannot have a metric of positive -intermediate curvature if the boundary is -convex, and some rigidity result holds if -intermediate curvature is nonnegative. This non-existence persists after performing a connected sum with an arbitrary manifold. These results generalize results of \cite{brendle,chenshuli,ChuKwongLee,Xu} to manifold with boundary.
Paper Structure (8 sections, 16 theorems, 74 equations)

This paper contains 8 sections, 16 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminary
  4. Proof of Theorem \ref{['main theorem 1 Introduction']}
  5. Nonexistence under connected sum
  6. Free boundary $\mu$-bubble
  7. Proof of Theorem \ref{['main theorem 2']}
  8. Connected sum in the boundary

Key Result

Theorem 1.1

Let $(M,\partial M)$ be a compact orientable $n$-dimensional manifold. Suppose that there exist nonzero $\alpha^{1},...,\alpha^{m}\in H^{1}(M,\mathbb{Z})$ such that $(\alpha^{1}\smile\cdots\smile \alpha^{m})|_{\partial M}\neq 0$. Then (1) If $2\leq n\leq 7$ and $1\leq m\leq n-1$, there is no Riemman

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 28 more