Open AccessThe balance between existence and nonexistence theorems in differential geometry
Author(s) -
Shihsuh Walter Wei
Publication year - 2001
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.32.2001.370
Subject(s) - mathematics , differential geometry , differential topology , pure mathematics , differential (mechanical device) , gauss , gaussian curvature , geometric analysis , balance (ability) , geometry and topology , harmonic map , nonlinear system , curvature , mathematical analysis , sectional curvature , geometry , topology (electrical circuits) , differential equation , ricci flat manifold , combinatorics , scalar curvature , physics , ordinary differential equation , medicine , differential algebraic equation , quantum mechanics , physical medicine and rehabilitation , thermodynamics
We discuss the delicate balance between existence and nonexistence theorems in differential geometry. Studying their interplay yields some information about $ p $-harmonic maps, $ p $-SSU manifolds, geometric $ k_p $-connected manifolds, minimal hypersurfaces and Gauss maps, and manifolds admitting essential positive supersolutions of certain nonlinear PDE. As an application of the theory developed, we obtain a topological theorem for minimal submanifolds in complete manifolds with nonpositive sectional curvature