Preserving positive intermediate curvature

TL;DR

This paper investigates the preservation of positive intermediate curvature on compact manifolds, establishing conditions for interpolating metrics and proving non-existence results for certain boundary configurations.

Contribution

It introduces a method to interpolate between metrics with positive m-intermediate curvature and proves non-existence results for partial torical bands with specific boundary convexity.

Findings

Existence of smooth interpolating metrics with positive m-intermediate curvature

Non-existence of partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries

Obstructions on partial tori for positive m-intermediate curvature

Abstract

Consider a compact manifold $N$ (with or without boundary) of dimension $n$. Positive $m$-intermediate curvature interpolates between positive Ricci curvature ($m = 1$) and positive scalar curvature ($m = n-1$), and it is obstructed on partial tori $N^n = M^{n-m} \times \mathbb{T}^m$. Given Riemannian metrics $g, \bar{g}$ on $(N, \partial N)$ with positive $m$-intermediate curvature and $m$-positive difference $h_g - h_{\bar{g}}$ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive $m$-intermediate curvature interpolating between $g$ and $\bar{g}$. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive $m$-intermediate curvature and strictly $m$-convex boundaries.