TY - JOUR
T1 - Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
AU - Marques, Fernando C.
AU - Neves, André
N1 - Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
PY - 2017/8/1
Y1 - 2017/8/1
N2 - In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
AB - In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
UR - http://www.scopus.com/inward/record.url?scp=85010727673&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85010727673&partnerID=8YFLogxK
U2 - 10.1007/s00222-017-0716-6
DO - 10.1007/s00222-017-0716-6
M3 - Article
AN - SCOPUS:85010727673
SN - 0020-9910
VL - 209
SP - 577
EP - 616
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 2
ER -