On 1-Harmonic Functions

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over$\mathbb{R}$; and every 7-dimensional$SO(2)\times SO(6)$-invariant absolutely area-minimizing integral current in$\mathbb{R}^8$is real analytic. The assumption on the$SO(2) \times SO(6)$-invariance cannot be removed, due to the first counter-example in$\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 pp-Subharmonic Functions, p>1p >1 On Rn\mathbb{R}^n 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 $p$-subharmonicity, subsolutions to the $p$-Laplace equation, uniqueness, existence, isometric immersions in multiple settings. In particular, we show that a convex function on $\mathbb{R}^n$ is a $p$-subharmonic function, for every $p > 1$, and a $C^2$ convex function on a Riemannian manifold is a $p$-subharmonic function $f$, for every $p > 1\, .$ We also show that a $C^2$ convex function which is a submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p \ge 1\, .$ This result is sharp. As further applications, via function growth estimates in $p$-harmonic geometry, we prove that every $p$-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for $p > 1$. In particular, every $L^q$, nonnegative, convex function of class $C^2$ on a complete noncompact Riemannian manifold is constant for $q > p -1 > 0\, .$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

nn-Harmonicity, Minimality, Conformality and Cohomology

By studying cohomology classes that are related with $n$-harmonic morphisms and $F$-harmonic maps, we augment and extend several results on $F$-harmonic maps, harmonic maps in [1, 3, 14], $p$-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and $n$-harmonic morphisms which are submersions. The results, for example Theorem 3.2 obtained by utilizing the $n$-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 11-harmonic Equation and The Inverse Mean Curvature Flow

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equations (1.1) on a domain in a Riemannian $n$-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 $p$-subharmonic functions of constant $p$-tension field, $p \ge n$, 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