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Conformally weighted Einstein manifolds: the uniqueness problem

Miguel Brozos-Vázquez, Eduardo García-Río, Diego Mojón-Álvarez

TL;DR

The paper addresses the non-uniqueness problem for weighted Einstein SMMS within a weighted conformal class by deriving both local and global classifications. It shows that the conformal factor satisfies a generalized Obata equation, forcing a warped-product structure locally and enabling reduction to ODEs for the base and fiber geometries. The main results classify complete SMMS admitting two weighted Einstein representatives: either they are weighted space forms or specific warped products with a detailed fiber structure, and in the compact setting they reduce to weighted spheres. These findings extend known results from the unweighted ($f$ constant) setting to the broader weighted context, with implications for weighted Yamabe-type problems and rigidity phenomena in weighted geometry.

Abstract

We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped products. Secondly, a global classification result is obtained when one of the underlying metrics is complete, showing that either it is a weighted space form, a special Einstein warped product, or a specific family of quasi-Einstein warped products. As a consequence, it must be a weighted sphere in the compact case.

Conformally weighted Einstein manifolds: the uniqueness problem

TL;DR

The paper addresses the non-uniqueness problem for weighted Einstein SMMS within a weighted conformal class by deriving both local and global classifications. It shows that the conformal factor satisfies a generalized Obata equation, forcing a warped-product structure locally and enabling reduction to ODEs for the base and fiber geometries. The main results classify complete SMMS admitting two weighted Einstein representatives: either they are weighted space forms or specific warped products with a detailed fiber structure, and in the compact setting they reduce to weighted spheres. These findings extend known results from the unweighted ( constant) setting to the broader weighted context, with implications for weighted Yamabe-type problems and rigidity phenomena in weighted geometry.

Abstract

We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped products. Secondly, a global classification result is obtained when one of the underlying metrics is complete, showing that either it is a weighted space form, a special Einstein warped product, or a specific family of quasi-Einstein warped products. As a consequence, it must be a weighted sphere in the compact case.

Paper Structure

This paper contains 9 sections, 8 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: weighted Einstein condition and conformal classes
  3. Weighted Einstein SMMSs
  4. Conformal classes
  5. Regularity of solutions
  6. Weighted space forms
  7. Conformal weighted Einstein SMMs: local study
  8. Remarks and examples
  9. The complete case

Key Result

Theorem 1.1

Let $(M^n,g,f,m,\mu)$ be a weighted Einstein SMMS, with $P_f^m=\lambda g$, and such that there exists a conformally equivalent weighted Einstein SMMS $(M^n,\widehat{g},\widehat{f},m,\mu)$, with $\widehat{P}_f^m=\widehat{\lambda} \widehat{g}$. Then, on a neighborhood of each regular point of the conf

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Example 2.3: $m$-weighted $n$-sphere $(\mathbb{S}^n(2\lambda),g_{\mathbb{S}}^{2\lambda},f_m,m,\mu)$
  • Example 2.4: $m$-weighted $n$-Euclidean space $(\mathbb{R}^n,g_\mathbb{E},f_m,m,\mu)$
  • Example 2.5: $m$-weighted $n$-hyperbolic space $(\mathbb{H}^n(2\lambda),g_{\mathbb{H}}^{2\lambda},f_m,m,\mu)$
  • ...and 14 more