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Upper bound preservation of the total scalar curvature in a conformal class

Shota Hamanaka

TL;DR

The paper addresses whether the upper bound on the total scalar curvature is preserved under $C^{0}$-limits within a fixed conformal class on a closed manifold, with the answer depending on the Yamabe constant $Y(M,g_0)$. The author uses the volume-preserving normalized Yamabe flow $g(t)=u(t)^{4/(n-2)} g_0$ and subsequential flow convergence from approximating metrics $g_i = u_i^{4/(n-2)} g_0$ to prove $\int_M R(g) \, dvol_g \le \kappa$ in the limit; this is treated separately for $Y(M,g_0)\le 0$ and $Y(M,g_0) > 0$, the latter requiring an additional lower bound on $R(g)$. The main results show that within a fixed conformal class, the upper bound set for total scalar curvature is $C^{0}$-closed when $Y(M,g_0)\le 0$, and, for $Y(M,g_0)>0$, the intersection with metrics having a lower scalar curvature bound is also $C^{0}$-closed under suitable assumptions; a weaker assumption still yields the same conclusion. Overall, the work integrates flow techniques with conformal geometry to establish stability of total scalar curvature bounds under $C^{0}$-limits, extending previous closure results in a conformal context.

Abstract

We show that in an arbitrarily fixed conformal class on a closed manifold, the upper bound condition of the total scalar curvature is $C^{0}$-closed if its Yamabe constant is nonpositive. Moreover, we show that if a conformal class on a closed manifold has positive Yamabe constant, then the intersection of such conformal class and the space of all Riemannian metrics, whose scalar curvatures are bounded from below as well as total scalar curvatures are bounded from above is $C^{0}$-closed in the space of all Riemannian metrics.

Upper bound preservation of the total scalar curvature in a conformal class

TL;DR

The paper addresses whether the upper bound on the total scalar curvature is preserved under -limits within a fixed conformal class on a closed manifold, with the answer depending on the Yamabe constant . The author uses the volume-preserving normalized Yamabe flow and subsequential flow convergence from approximating metrics to prove in the limit; this is treated separately for and , the latter requiring an additional lower bound on . The main results show that within a fixed conformal class, the upper bound set for total scalar curvature is -closed when , and, for , the intersection with metrics having a lower scalar curvature bound is also -closed under suitable assumptions; a weaker assumption still yields the same conclusion. Overall, the work integrates flow techniques with conformal geometry to establish stability of total scalar curvature bounds under -limits, extending previous closure results in a conformal context.

Abstract

We show that in an arbitrarily fixed conformal class on a closed manifold, the upper bound condition of the total scalar curvature is -closed if its Yamabe constant is nonpositive. Moreover, we show that if a conformal class on a closed manifold has positive Yamabe constant, then the intersection of such conformal class and the space of all Riemannian metrics, whose scalar curvatures are bounded from below as well as total scalar curvatures are bounded from above is -closed in the space of all Riemannian metrics.
Paper Structure (4 sections, 10 theorems, 68 equations)

This paper contains 4 sections, 10 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Proof of Main Theorem \ref{['main']}
  3. Proof of Main Theorem \ref{['main2']}
  4. Appendix

Key Result

Corollary 1.1

Let $M^{n}$ be a closed manifold of dimension $n \ge 2$ and $g_{0}$ a $C^{2}$-Riemannian metric on $M.$ Let $\mathcal{M}$ be the space of all $C^{2}$-Riemannian metrics on $M.$ Then, for any continuous function $\sigma \in C^{0}(M)$ and constant $\kappa \in \mathbb{R},$ the space when $Y(M, g_{0}) > 0,$ and the space when $Y(M, g_{0}) \le 0$ are both closed in $\mathcal{M}$ with respect to $C^{0

Theorems & Definitions (25)

  • Corollary 1.1
  • Proposition 2.1: carron2021convergence
  • proof
  • Remark 2.1
  • Theorem 2.1: chow1992yamabeschwetlick2003convergencebrendle2005convergence, cf. carron2021convergence
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.2: brendle2005convergence
  • proof
  • Proposition 2.3: brendle2005convergence
  • ...and 15 more