Optimal bounds for ancient caloric functions
Tobias Colding, William P. Minicozzi
Abstract
For any manifold with polynomial volume growth, we show that the dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau’s 1974 conjecture about polynomial growth harmonic functions holds.
Topics & Concepts
Dimension (graph theory)MathematicsBounded functionPolynomialConjectureCaloric theorySpace (punctuation)Pure mathematicsManifold (fluid mechanics)Mathematical analysisPhysicsComputer scienceOperating systemMechanical engineeringEngineeringThermodynamicsGeometric Analysis and Curvature FlowsMeromorphic and Entire FunctionsNonlinear Partial Differential Equations