Prescribing the mixed scalar curvature of a foliation

Vladimir Rovenski, Leonid Zelenko

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We introduce the flow of metrics on a foliated Riemannian manifold .M; g/, whose velocity along the orthogonal (to the foliation F) distribution D is proportional to the mixed scalar curvature, Scalmix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scalmix (e.g., positive/negative or constant)? If the mean curvature vector of D is leaf-wise conservative, then its potential function obeys the nonlinear heat equation (Formula Presented) with a leaf-wise constant Φ and known functions βD ≥ 0 and ψFi ≥ 0. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrödinger operator) the flow of metrics admits a unique global solution, whose Scalmix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scalmix is negative, positive, or negative constant.

Original languageEnglish
Title of host publicationGeometry and its Applications
EditorsPawel Walczak, Vladimir Rovenski
PublisherSpringer New York LLC
Number of pages41
ISBN (Electronic)9783319046747
StatePublished - 2014
Event2nd International workshop Geometry and Symbolic Computation, 2013 - Haifa, Israel
Duration: 15 May 201318 May 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


Conference2nd International workshop Geometry and Symbolic Computation, 2013

Bibliographical note

Publisher Copyright:
© Springer International Publishing Switzerland 2014.


  • Conformal
  • Foliation
  • Mean curvature vector
  • Mixed scalar curvature
  • Parabolic PDE
  • Riemannian metric
  • Schrödinger operator
  • Twisted product

ASJC Scopus subject areas

  • General Mathematics


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