TY - JOUR
T1 - Minimal hypersurfaces with bounded index
AU - Chodosh, Otis
AU - Ketover, Daniel
AU - Maximo, Davi
N1 - Funding Information:
We are grateful to Dave Gabai for several illuminating conversations. We thank Nick Edelen for several useful comments and Alessandro Carlotto for pointing out a mistaken reference in an earlier version of Corollary . We would like to thank the referee for several useful suggestions. O.C. would like to acknowledge Jacob Bernstein for a helpful discussion regarding removable singularity results as well as Sander Kupers for kindly answering some questions about topology. He was partially supported by an NSF Graduate Research Fellowship DGE-1147470 during the time which most of this work was undertaken, and also acknowledges support by the EPSRC Programme Grant, Singularities of Geometric Partial Differential Equations number EP/K00865X/1. D.K. was partially supported by an NSF Postdoctoral Fellowship DMS-1401996. D.M. would like to acknowledge Richard Schoen for several helpful conversations. He was partially supported by NSF grant DMS-1512574 during the completion of this work.
Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold (Mn, g) , 3 ≤ n≤ 7 , can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follow from our local picture.
AB - We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold (Mn, g) , 3 ≤ n≤ 7 , can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follow from our local picture.
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U2 - 10.1007/s00222-017-0717-5
DO - 10.1007/s00222-017-0717-5
M3 - Article
AN - SCOPUS:85012917646
SN - 0020-9910
VL - 209
SP - 617
EP - 664
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -