## Abstract

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, Scal_{mix}. 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 Scal_{mix} (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 β_{D} ≥ 0 and ψ^{F}_{i} ≥ 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 Scal_{mix} converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a D-conformal to g metric, whose Scal_{mix} is negative, positive, or negative constant.

Original language | English |
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Title of host publication | Geometry and its Applications |

Editors | Pawel Walczak, Vladimir Rovenski |

Publisher | Springer New York LLC |

Pages | 83-123 |

Number of pages | 41 |

ISBN (Electronic) | 9783319046747 |

DOIs | |

State | Published - 2014 |

Event | 2nd International workshop Geometry and Symbolic Computation, 2013 - Haifa, Israel Duration: 15 May 2013 → 18 May 2013 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 72 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | 2nd International workshop Geometry and Symbolic Computation, 2013 |
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Country/Territory | Israel |

City | Haifa |

Period | 15/05/13 → 18/05/13 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2014.

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics