Scholarlink

<p>Is a sequence of Riemannian manifolds with positive scalar curvature, satisfying some conditions to keep the sequence reasonable, compact? What topology should one use for the convergence and what is the regularity of the limit space? In this paper we explore these questions by studying the case of a sequence of Riemannian manifolds which are conformal to the <i>n</i>-dimensional round sphere. We are able to show that the sequence of conformal factors are compact in several analytic senses and are able to establish <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> convergence away from a singular set of small volume in a similar fashion as Dong (Québec <b>48</b>(2) 427–451 (2024)). Under a bound on the total scalar curvature we are able to show that the limit conformal factor has weak positive scalar curvature in the sense of weakly solving the conformal positive scalar curvature equation.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On the scalar curvature compactness conjecture in the conformal case

  • 期刊论文

  • 发表日期：

    2025年12月4日

查看全文
  • Brian Allen,
  • Wenchuan Tian,
  • Changliang Wang

摘要

Is a sequence of Riemannian manifolds with positive scalar curvature, satisfying some conditions to keep the sequence reasonable, compact? What topology should one use for the convergence and what is the regularity of the limit space? In this paper we explore these questions by studying the case of a sequence of Riemannian manifolds which are conformal to the n-dimensional round sphere. We are able to show that the sequence of conformal factors are compact in several analytic senses and are able to establish \(C^0\) C 0 convergence away from a singular set of small volume in a similar fashion as Dong (Québec 48(2) 427–451 (2024)). Under a bound on the total scalar curvature we are able to show that the limit conformal factor has weak positive scalar curvature in the sense of weakly solving the conformal positive scalar curvature equation.