arXiv research

Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.

We classify homothetical surfaces with constant mean curvature in hyperbolic space.

Study on minimal surfaces in a 3D space with 2m-norm.

A homothetical surface arises as a graph of a function $z = \varphi_1(v_1) \varphi_2(v_2)$. In this paper, we study the homothetical surfaces in three dimensional psuedo-Galilean space$\left(\mathbb{G}_3^1\right)$ satisfying the conditions $Δ^{II}\textbf{x}_i=λ_i\textbf{x}_i,$ where $Δ^{II}$ is the Laplacian with respe…

Proves uniqueness of catenoid-like shapes in a ball.

The paper analyzes self-similar solutions for mean curvature flow in 3D.

Symmetry groups help define solitons in curved spaces.

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to non degenerate surfaces in Lorentz-Minkowski space.

New rigidity result for hyperbolic surfaces based on curve lengths.

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

Solves geodesic completeness on pseudo-homothetic Lie group.

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…

Mathematically, a homothetic function is a function of the form $f({\bf x})=F(h(x_1,...,x_n))$, where $h$ is a homogeneous function of any degree $d\ne 0$ and $F$ is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneo…

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…

We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.

The paper explores conditions for homothetic Killing vectors on spacetime hypersurfaces.

The study finds minimal distortion embeddings of surfaces into small domains.

The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliation by curves.

In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.

We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.

In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(α,β)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of $(α,β)$ spaces under certain c…

Study proves conditions for translating solitons to be planar.

We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of …

In this paper, we completely classify the homothetical hypersurfaces having null Gauss-Kronocker curvature in a Euclidean (n+1)-space. Several applications to the production functions in economics are also given.

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward …

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

Study on solitons in deformed Kenmotsu manifolds with specific vector fields.

Study proves existence of expanding solutions for multiphase surfaces with regular junctions.

The paper classifies vertices in planar polygons formed by convex domains.

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large…

Flat holonomies imply homotheticity in certain curved spaces.

In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat $(α,β)$-space of non-Randers type in dimension $n\ge 3$, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fiel…

Study on surfaces with constant anisotropic mean curvature in 3D space.

This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's monotonicity formula. Examples of homothetically shrinking …

Proves stability of cone-volume measure with nearly constant density.

In this paper, we provide conceptional explanations for the geodesic and Jacobi field correspondences for homothetic navigation, and then let them guide us to the shortcuts to some well known flag curvature and S-curvature formulas. They also help us directly see the local correspondence between isoparametric functions…

In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future …

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{…

Study solves overdetermined problem on compact surfaces.

In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when $M$ admits…

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…

Study on CPSRM from/to Kähler manifolds, deriving integrability and geodesic results.

Kähler cones over Sasakian manifolds are flat if projectively induced.

New proof of non-compact homothetic solitons for inverse mean curvature flow.

Study on Kähler-Ricci flow and conformal submersion singularity formation.

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $\mathcal{D}$-homothetic transformation is determined as a special gauge transformation. The $η$-Einstein manifold are defined, it is prove that their scalar curvature is a …