Abstract: We are interested in solving Liouville-type problems to explore constancy properties for maps or differential forms on Riemannian manifolds. Geometric structures on manifolds, the existence of constancy properties for maps or differential forms, and energy growth for maps or differential forms are intertwined. In this article, we concentrate on discovery of solutions to Liouville-type problems where manifolds are Euclidean spaces (i.e. flat Riemannian manifolds) and maps become real-valued functions. Liouville-type results of vanishing properties for functions are obtained. The original work in our research findings is to extend the q-energy for a function from finite in Lq space to infinite in non-Lq space by applying p-balanced technique where q = p = 2. Calculation skills such as Hölder's Inequality and Tests for Series have been used to evaluate limits and integrations for function energy. Calculation ideas and computational techniques for solving Liouville-type problems shown in this article, which are utilized in Euclidean spaces, can be universalized as a successful algorithm, which works for both maps and differential forms on Riemannian manifolds. This innovative algorithm has a far-reaching impact on research work of solving Liouville-type problems in the general settings involved with infinite energy. The p-balanced technique in this algorithm provides a clue to success on the road of q-energy extension from finite to infinite.
Abstract: The goal of this project is to investigate constant
properties (called the Liouville-type Problem) for a p-stable map
as a local or global minimum of a p-energy functional where
the domain is a Euclidean space and the target space is a
closed half-ellipsoid. The First and Second Variation Formulas
for a p-energy functional has been applied in the Calculus
Variation Method as computation techniques. Stokes’ Theorem,
Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and
the Bochner Formula as estimation techniques have been used to
estimate the lower bound and the upper bound of the derived
p-Harmonic Stability Inequality. One challenging point in this project
is to construct a family of variation maps such that the images
of variation maps must be guaranteed in a closed half-ellipsoid.
The other challenging point is to find a contradiction between the
lower bound and the upper bound in an analysis of p-Harmonic
Stability Inequality when a p-energy minimizing map is not constant.
Therefore, the possibility of a non-constant p-energy minimizing
map has been ruled out and the constant property for a p-energy
minimizing map has been obtained. Our research finding is to explore
the constant property for a p-stable map from a Euclidean space into
a closed half-ellipsoid in a certain range of p. The certain range of
p is determined by the dimension values of a Euclidean space (the
domain) and an ellipsoid (the target space). The certain range of p
is also bounded by the curvature values on an ellipsoid (that is, the
ratio of the longest axis to the shortest axis). Regarding Liouville-type
results for a p-stable map, our research finding on an ellipsoid is a
generalization of mathematicians’ results on a sphere. Our result is
also an extension of mathematicians’ Liouville-type results from a
special ellipsoid with only one parameter to any ellipsoid with (n+1)
parameters in the general setting.