19 research outputs found
On 1-Harmonic Functions
Characterizations of entire subsolutions for the 1-harmonic equation of a
constant 1\mathbb{R}SO(2)\times SO(6)\mathbb{R}^8SO(2) \times SO(6)\mathbb{R}^8$, proved by Bombieri, De Girogi and
Giusti.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Convex Functions are -Subharmonic Functions, On with Applications
In this paper we discuss convexity, its average principle, an extrinsic
average variational method in the Calculus of Variations, an average method in
Partial Differential Equations, a link of convexity to -subharmonicity,
subsolutions to the -Laplace equation, uniqueness, existence, isometric
immersions in multiple settings. In particular, we show that a convex function
on is a -subharmonic function, for every , and a
convex function on a Riemannian manifold is a -subharmonic function , for
every We also show that a convex function which is a
submersion on a Riemannian manifold is a -subharmonic function, for every This result is sharp. As further applications, via function growth
estimates in -harmonic geometry, we prove that every -balanced
nonnegative convex function on a complete noncompact Riemannian manifold
is constant for . In particular, every , nonnegative, convex
function of class on a complete noncompact Riemannian manifold is
constant for Comment: 12 pages, 1 table, to appear in Lecture Notes of Seminario
Interdisciplinare di Matematica Vol 16 (2023). arXiv admin note: text overlap
with arXiv:2104.0512
-Harmonicity, Minimality, Conformality and Cohomology
By studying cohomology classes that are related with -harmonic morphisms
and -harmonic maps, we augment and extend several results on -harmonic
maps, harmonic maps in [1, 3, 14], -harmonic morphisms in [17], and also
revisit our previous results in [9, 10, 21] on Riemannian submersions and
-harmonic morphisms which are submersions. The results, for example Theorem
3.2 obtained by utilizing the -conservation law (2.6), are sharp.Comment: 10 pages, arXiv:2302.14019v[1] 10 pages, arXiv:2302.14019v[1],
arXiv:2302.14019v[2], to appear in Tamkang Journal of Mathematic
Discovering Geometric and Topological Properties of Ellipsoids by Curvatures
Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting.
Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics.
Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), and Department of Mathematics (Central Michigan University), between January 2014 and February 2015.
Methodology: Calculating curvatures of a surface is now at the threshold of a better understanding regarding geometric, topological and physical properties on a surface. In order to calculate various curvatures, we demonstrate the way to compute the second fundamental form associated with curvatures by extending the calculation method from spheres to ellipsoids.
Results: Just as curvatures of a sphere are determined by its radius, curvatures of an ellipsoid are determined by its longest axis and its shortest axis. On an ellipsoid, the value of the ratio of its longest axis to its shortest axis is also a critical index to characterize its geometric, topological and physical behaviors.
Conclusion: Our results on ellipsoids are extensions or generalizations of results of Lawson-Simons, Wei, and Simons on spheres, and Kobayashi-Ohnita-Takeuchi on an ellipsoid with “one variable”. Methods and research findings in this paper can lead to future research on convex hyper-surfaces
Generalized -harmonic Equation and The Inverse Mean Curvature Flow
We introduce and study generalized -harmonic equations (1.1). Using some
ideas and techniques in studying -harmonic functions from [W1] (2007), and
in studying nonhomogeneous -harmonic functions on a cocompact set from [W2,
(9.1)] (2008), we find an analytic quantity in the generalized -harmonic
equations (1.1) on a domain in a Riemannian -manifold that affects the
behavior of weak solutions of (1.1), and establish its link with the geometry
of the domain. We obtain, as applications, some gradient bounds and
nonexistence results for the inverse mean curvature flow, Liouville theorems
for -subharmonic functions of constant -tension field, , and
nonexistence results for solutions of the initial value problem of inverse mean
curvature flow.Comment: 14 pages, to appear in Journal of Geometry and Physic