Open AccessBerger’s Inequality in the Presence of Upper Sectional Curvature Bound
Author(s) -
Gerasim Kokarev
Publication year - 2021
Publication title -
international mathematics research notices
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.757
H-Index - 76
eISSN - 1687-0247
pISSN - 1073-7928
DOI - 10.1093/imrn/rnab073
Subject(s) - mathematics , sectional curvature , upper and lower bounds , eigenvalues and eigenvectors , curvature , laplace transform , laplace operator , mathematical analysis , pure mathematics , conformal map , space (punctuation) , inequality , scalar curvature , geometry , linguistics , philosophy , physics , quantum mechanics
We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the 1st Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal submanifolds. In addition, we obtain explicit upper bounds for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.