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A remark on $Λ^2$-enlargeable manifolds

Guangxiang Su

TL;DR

This work extends the Λ^2-enlargeability framework to the locally constant near infinity setting and shows that such manifolds cannot admit complete metrics of positive scalar curvature, mirroring the classical GL83 obstruction. It then provides an alternative proof of Wang-Zhang's theorem on the generalized Geroch conjecture for complete spin manifolds, using coverings, contracting maps to spheres, and a deformation of twisted Dirac operators together with the Lichnerowicz formula and the Atiyah-Singer index theorem. The key mechanism is an index-theoretic contradiction arising from a nonzero degree map on a suitable covering, which rules out PSC in this noncompact context. Overall, the paper strengthens nonexistence results for PSC under enlargeability-type conditions and highlights deep links between spin geometry, index theory, and geometric analysis on noncompact manifolds.

Abstract

In this note, we consider the case that the condition ``constant near infinity" in the definition of $Λ^2$-enlargeable manifold replaced by the condition ``locally constant near infinity" and prove that $Λ^2$-enlargeable manifold in the current sense also can not carry a complete Riemannian metric of positive scalar curvature. As a consequence, we give another proof of Wang-Zhang's theorem on the generalized Geroch conjecture for complete spin manifolds.

A remark on $Λ^2$-enlargeable manifolds

TL;DR

This work extends the Λ^2-enlargeability framework to the locally constant near infinity setting and shows that such manifolds cannot admit complete metrics of positive scalar curvature, mirroring the classical GL83 obstruction. It then provides an alternative proof of Wang-Zhang's theorem on the generalized Geroch conjecture for complete spin manifolds, using coverings, contracting maps to spheres, and a deformation of twisted Dirac operators together with the Lichnerowicz formula and the Atiyah-Singer index theorem. The key mechanism is an index-theoretic contradiction arising from a nonzero degree map on a suitable covering, which rules out PSC in this noncompact context. Overall, the paper strengthens nonexistence results for PSC under enlargeability-type conditions and highlights deep links between spin geometry, index theory, and geometric analysis on noncompact manifolds.

Abstract

In this note, we consider the case that the condition ``constant near infinity" in the definition of -enlargeable manifold replaced by the condition ``locally constant near infinity" and prove that -enlargeable manifold in the current sense also can not carry a complete Riemannian metric of positive scalar curvature. As a consequence, we give another proof of Wang-Zhang's theorem on the generalized Geroch conjecture for complete spin manifolds.
Paper Structure (2 sections, 2 theorems, 24 equations)

This paper contains 2 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. A proof of Theorem \ref{['th1.3']}

Key Result

Theorem 1.3

A manifold $M$ without boundary which is $\Lambda^2$-enlargeable in the sense of Definition df1.2, can not carry a complete Riemannian metric of positive scalar curvature.

Theorems & Definitions (5)

  • Definition 1.1: GL83
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • proof