The mixed Yamabe problem for foliations

Vladimir Rovenski, Leonid Zelenko

Research output: Contribution to journalArticlepeer-review

Abstract

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the leafwise constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a harmonic foliation, we derive the leafwise elliptic equation and explore the corresponding nonlinear heat type equation on a closed manifold (leaf). Then we assume that a foliation is defined by an orientable fiber bundle, and use spectral parameters of certain Schrödinger operator to find solution, which is an attractor of the equation.

Original languageEnglish
Pages (from-to)503-533
Number of pages31
JournalEuropean Journal of Mathematics
Volume1
Issue number3
DOIs
StatePublished - 1 Sep 2015

Bibliographical note

Publisher Copyright:
© 2015, European Union.

Keywords

  • Attractor
  • Conformal
  • Foliation
  • Harmonic
  • Leafwise Schrödinger operator
  • Mixed scalar curvature
  • Parabolic equation
  • Riemannian metric

ASJC Scopus subject areas

  • Mathematics (all)

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