arXiv subjects

Sin-Hua Lai

Publications and source records attributed to Sin-Hua Lai.

4 recordsLinked to original sources

arXiv2025

The Classification of Rotationally symmetric hypersurfaces in the Heisenberg groups $H_{n}$

In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups $H_{n}$. In addition, we give a complete description of generating curves for rotationally symmetric hypersurfaces with constant $p$-mean curvature $H=c$ (including $H=0$) in the Heisenberg group $H_{n}$. We also establish the validity of Alexandrov's theorem for rotationally symmetric hypersurfaces in $H_n$.

math.DG↗

arXiv2025

On Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$

One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant $p$-mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant $p$-mean curvature and shed light on the geometric interpretation of the energy $E$ with a lower bound.

math.DG↗

arXiv2017

An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.

math.DG↗

arXiv2013

The fundamental theorems for curves and surfaces in 3d Heisenberg group

We study the local equivalence problems of curves and surfaces in three dimensional Heisenberg group via Cartans method of moving frames and Lie groups, and find a complete set of invariants for curves and surfaces. For surfaces, in terms of these invariants and their suitable derivatives, we also give a Gaussian curvature fromula of the metric induced from the adapted metric on Heisenberg group, and hence form a new formula for the Euler number of a closed surface.

math.DG↗