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On the Yamabe constants of product manifolds

Chanyoung Sung

TL;DR

The paper develops a fiberwise spherical symmetrization method to compare Yamabe constants on warped product manifolds and to establish the existence of radially symmetric Yamabe minimizers on products with round spheres. It proves a sharp warped-product Yamabe inequality, characterizes the equality case, and shows that minimizers can be chosen invariant under enlarged symmetry groups, yielding symmetric minimizers on $N\times\prod S^{m_i}$. Applications include a lower bound for $Y(S^1\times M)$ in terms of $Y(S^{m+1})$, and a new route to the 3D optimal-volume sphere theorem, with implications for Einstein metrics and diffeomorphism classifications. Overall, the results advance product-formula techniques in the Yamabe problem and provide concrete symmetry-adept minimizers and corollaries for low-dimensional geometries.

Abstract

By using the fiberwise spherical symmetrization we give a comparison theorem of Yamabe constants on warped products and prove the existence of radially-symmetric Yamabe minimizers on Riemannian manifolds given by products with round spheres.

On the Yamabe constants of product manifolds

TL;DR

The paper develops a fiberwise spherical symmetrization method to compare Yamabe constants on warped product manifolds and to establish the existence of radially symmetric Yamabe minimizers on products with round spheres. It proves a sharp warped-product Yamabe inequality, characterizes the equality case, and shows that minimizers can be chosen invariant under enlarged symmetry groups, yielding symmetric minimizers on . Applications include a lower bound for in terms of , and a new route to the 3D optimal-volume sphere theorem, with implications for Einstein metrics and diffeomorphism classifications. Overall, the results advance product-formula techniques in the Yamabe problem and provide concrete symmetry-adept minimizers and corollaries for low-dimensional geometries.

Abstract

By using the fiberwise spherical symmetrization we give a comparison theorem of Yamabe constants on warped products and prove the existence of radially-symmetric Yamabe minimizers on Riemannian manifolds given by products with round spheres.

Paper Structure

This paper contains 9 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and Notations
  4. Convergence of minimizing sequence
  5. Fiberwise symmetrization
  6. Yamabe constant comparison and Proof of Theorem \ref{['Yoon']}
  7. Symmetric minimizers and Proof of Theorem \ref{['newsfactory']}
  8. Yamabe invariant of $S^1\times M$
  9. Optimal volume sphere theorem

Key Result

Theorem 1.1

Let $(M^m,g)$ be a smooth closed Riemannian $m$-manifold with positive scalar curvature. Then there exists a constant $K>0$ depending only on $(M,g)$ such that for any smooth closed manifold $N$ of dimension $n$,

Theorems & Definitions (20)

  • Theorem 1.1: Petean Petean-1
  • Theorem 1.2: Ammann, Dahl, and Humbert ADH
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • Remark 3.1
  • Lemma 4.1
  • ...and 10 more