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Hsiao-Fan Liu

Publications and source records attributed to Hsiao-Fan Liu.

10 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↗
arXiv2022

A characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$

In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature $K=-1$. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group $H_{1}$, we show in this paper that the existence of a constant $p$-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of {\bf Liénard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant $p$-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant $p$-mean curvature surfaces and classify further all constant $p$-mean curvature surfaces. In Section 9, we provide an approach to construct $p$-minimal surfaces. It turns out that, in some sense, generic $p$-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [3] (or see [7,8]).

math.DG↗
arXiv2021

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

The paper deals with Sturm-Liouville-type operators with frozen argument of the form $\ell y:=-y''(x)+q(x)y(a),$ $y^{(α)}(0)=y^{(β)}(1)=0,$ where $α,β\in\{0,1\}$ and $a\in[0,1]$ is an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the potential $q(x)$ from the spectrum of the operator $\ell.$ Our goal is two-fold. Firstly, we establish a deep connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first and the second kinds. This connection gives a new perspective method for solving the inverse problem. In particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient description of iso-spectral potentials in the space of complex-valued integrable functions.

math.SP↗
arXiv2019

Geometric Algorithm of Schrödinger Flow on a Sphere

We construct the solution to the periodic Cauchy problem of the Schrödinger flow on the sphere. Such construction of solutions is formulated explicitly and therefore a geometric algorithm of solving this periodic Cauchy problem follows. Theoretical and experimental results will be discussed.

math.DG↗
arXiv2018

Star Mean Curvature Flow on 3 manifolds and its Bäcklund Transformations

The Hodge star mean curvature flow on a 3-dimensional Riemannian or pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in geometric analysis. A curve flow is integrable if the local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we show that this flow on $\mathbb{S}^3$ and $\mathbb{H}^3$ are integrable, and describe algebraically explicit solutions to such curve flows. The Cauchy problem of the curve flows on $\mathbb{S}^3$ and $\mathbb{H}^3$ and its Bäcklund transformations follow from this construction.

math.DG↗
arXiv2017

Graphic Enumerations and Discrete Painlevé Equations via Random Matrix Models

We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with the quartic matrix model satisfy the discrete type I Painlevé equation. Through the use of generating function techniques, we show that the planar contribution to the free energy is controlled by the Catalan numbers. We also develop a new systematic scheme of calculating higher-genus contributions to the topological expansion of the free energy of matrix models. It is important that our exact solutions are valid for finite-$N$ matrix models and no continuous limits are taken within our approach. To show the advantages of our approach, we provide new results of the topological expansion of the free energy for the finite-$N$ cubic matrix model.

hep-th↗
arXiv2017

Anatomy of a q-generalization of the Laguerre/Hermite Orthogonal Polynomials

We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of quadratic relation is illustrated by comparing two ways of calculating recursive coefficients. Finally, we derive a q-deformed version of the Toda equations for both q-Laguerre/Hermite ensembles.

nlin.SI↗
arXiv2017

$N$-soliton formula and blowup result of the Wadati-Konno-Ichikawa equation

We formulate the $N$ soliton solution of the Wadati-Konno-Ichikawa equation that is determined by purely algebraic equations. Derivation is based on the matrix Riemann-Hilbert problem. We give examples of one soliton solution that include smooth soliton, bursting soliton, and loop type soliton. In addition, we give an explicit example for two soliton solution that blows up in a finite time.

nlin.SI↗