Graphical mean curvature flow with bounded bi-Ricci curvature

verfasst von

Renan Assimos, Andreas Savas-Halilaj, Knut Smoczyk

Abstract

We consider the graphical mean curvature flow of strictly area decreasing maps f: M→ N, where M is a compact Riemannian manifold of dimension m> 1 and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature BRic

_M of M is bounded from below by the sectional curvature σ

_N of N. In addition, we obtain smooth convergence to a minimal map if Ric

_M≥ sup { 0 , sup

_Nσ

_N}. These results significantly improve known results on the graphical mean curvature flow in codimension 2.

Organisationseinheit(en)

Institut für Differentialgeometrie
Riemann Center for Geometry and Physics

Externe Organisation(en)

University of Ioannina

Typ

Artikel

Journal

Calculus of Variations and Partial Differential Equations

Band

Anzahl der Seiten

ISSN

0944-2669

Publikationsdatum

01.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

ASJC Scopus Sachgebiete

Analysis, Angewandte Mathematik

Elektronische Version(en)

https://doi.org/10.48550/arXiv.2201.05523 (Zugang: Offen)
https://doi.org/10.1007/s00526-022-02369-3 (Zugang: Geschlossen)

BibTeX

@article{b72d536ec17042be9027543a14a01433,
title = "Graphical mean curvature flow with bounded bi-Ricci curvature",
abstract = "We consider the graphical mean curvature flow of strictly area decreasing maps f: M→ N, where M is a compact Riemannian manifold of dimension m> 1 and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature BRic M of M is bounded from below by the sectional curvature σ N of N. In addition, we obtain smooth convergence to a minimal map if Ric M≥ sup { 0 , sup Nσ N}. These results significantly improve known results on the graphical mean curvature flow in codimension 2.",
author = "Renan Assimos and Andreas Savas-Halilaj and Knut Smoczyk",
note = "Funding Information: The second author is supported by HFRI: Grant 133, and the third by DFG SM 78/7-1.",
year = "2023",
month = jan,
doi = "10.48550/arXiv.2201.05523",
language = "English",
volume = "62",
journal = "Calculus of Variations and Partial Differential Equations",
issn = "0944-2669",
publisher = "Springer New York",
number = "1",
}