Searcharxiv⌕ Search

arXiv subjects

Doan The Hieu

Publications and source records attributed to Doan The Hieu.

15 recordsLinked to original sources
arXiv2023

Complete $λ$-submanifolds in Gauss spaces

In this paper, we study $λ$-submanifolds of arbitrary codimensions in Gauss spaces. These submanifolds can be seen as natural generalizations of self-shrinker and $λ$-hypersurfaces. Using a divergence type theorem and some Simons' type identities, we prove some halfspace type theorems and gap theorems for complete proper $λ$-submanifolds. These generalized our as well as the others' results for self-shrinker or $λ$-hypersurfaces to $λ$-submanifolds.

math.DG↗
arXiv2018

A simple proof for Bernstein type theorems in Gauss space

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $λ$-hypersurfaces (\cite{ chwe2}) follow immediately as consequences.

math.DG↗
arXiv2017

Zero $f$-mean curvature surfaces of revolution in the Lorentzian product $\Bbb G^2\times\Bbb R_1$

We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\Bbb G^2\times\Bbb R_1,$ the Lorentz-Minkowski 3-space $\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\frac 1{2π}e^{-\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a horizontal plane or a spacelike $f$-Catenoid. For the timelike case, a timelike $f$-minimal surface is either a vertical plane containing $z$-axis, the cylinder $x^2+y^2=1,$ or a timelike $f$-Catenoid. Spacelike and timelike $f$-Catenoids are new examples of $f$-minimal surfaces in $\Bbb G^2\times \Bbb R_1.$

math.DG↗
arXiv2013

The classification of constant weighted curvature curves in the plane with a log-linear density

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^y.$

math.DG↗
arXiv2011

$HS_{r}$-valued Gauss maps and umbilic spacelike surfaces of codimension two

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umbilic spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$

math.DG↗
arXiv2010

On positive definite preserving linear transformations of rank $r$ on real symmetric matrices

We study on what conditions on $B_k,$ \ a linear transformation of rank $r$ \label{form} T(A)=\sum_{k=1}^r\tr(AB_k)U_k where $U_k,\ k=1,2,..., r$ are linear independent and all positive definite; is positive definite preserving. We give some first results for this question. For the case of rank one and two, the necessary and sufficient conditions are given. We also give some sufficient conditions for the case of rank $r.$

math.OA↗
arXiv2010

Some calibrated surfaces in manifolds with density

Hyperplanes, hyperspheres and hypercylinders in $\Bbb R^n$ with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.

math.DG↗
arXiv2009

Ruled minimal surfaces in $\Bbb R^3$ with density $e^z$

We classify ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is also proved that all translation minimal surfaces are ruled.

math.DG↗
arXiv2008

On the Four Vertex Theorem on planes with radial density $e^{ϕ(r)}$

It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density.

math.DG↗
arXiv2005

ADM submanifolds, SL normal bundles examples

It is showed that many examples of AMD submanifolds of higher dimensions come from SL normal bundles. A symmetry property of SL submanifolds and Björling type problem for SL normal bundles are also mentioned.

math.DG↗