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Guofang Wang

Publications and source records attributed to Guofang Wang.

At least 55 records · Page 3Linked to original sources
arXiv2021

Stable capillary hypersurfaces supported on a horosphere in the hyperbolic space

In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.

math.DG↗
arXiv2021

Meshy soliton structures for (2+1)-dimensional integrable systems and interactions

In this letter, we construct new meshy soliton structures by using two concrete (2+1)-dimensional integrable systems. The explicit expressions based on corresponding Cole-Hopf type transformations are obtained. Constraint equation ft+\sum_{j=1}^{N} h_j(y)f_{jx} = 0 shows that these meshy soliton structures can be linear or parabolic. Interaction between meshy soliton structure and Lump structure are also revealed.

nlin.SI↗
arXiv2020

The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in ${\mathbb S}^4$

In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into ${\mathbb S}^4$ [19]. This system is an integrable system which is a natural generalization of sinh-Gordon equation [18]. By exploring a refined blow-up analysis in the bubble domain, we prove that the blow-up values are multiple of $8π$, which generalizes the previous results proved in \cite{Spruck, OS, Jost-Wang-Ye-Zhou, Jevnikar-Wei-Yang} for the sinh-Gordon equation. Let $(u_k^1,u_k^2, u^3_k)$ be a sequence of solutions of \begin{align*} -Δu^1&=e^{u^1}-e^{u^3},\\ -Δu^2&=e^{u^2}-e^{u^3},\\ -Δu^3&=-\frac{1}{2}e^{u^1}-\frac{1}{2}e^{u^2}+ e^{u^3},\\ u^1+u^2+2u^3&=0, \end{align*} in $B_1(0)$, which has a uniformly bounded energy in $B_1(0)$, a uniformly bounded oscillation on $\partial B_1(0)$ and blows up at an isolated blow-up point $\{0\}$, then the local masses $(σ_1,σ_2, σ_3) \not = 0$ satisfy \begin{align*} \begin{array}{rcl} σ_1&=&m_1(m_1+3)+m_2(m_2-1)\\ σ_2&=& m_1(m_1-1)+m_2(m_2+3)\\ σ_3 &=& m_1(m_1-1)+m_2(m_2-1) \end{array} \, \qquad \hbox { for some } \begin{array} {l} (m_1, m_2)\in {\mathbb Z} \hbox { with }\\ m_1, m_2= 0 \hbox{ or } 1 \hbox{ mod } 4,\\ m_1, m_2 = 2 \hbox { or } 3\hbox { mod } 4. \end{array} \end{align*} Here the local mass is defined by $ σ_i:=\frac 1{2π}\lim_{δ\to 0}\lim_{k\to\infty}\int_{B_δ(0)}e^{u_k^i}dx.$

math.AP↗
arXiv2020

A mean curvature type flow with capillary boundary in a unit ball

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show that it has the longtime existence and subconverges to spherical caps. As an application, we solve an isoperimetric problem for hypersurfaces with capillary boundary.

math.DG↗
arXiv2020

Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

math.DG↗
arXiv2020

Lagrangian surfaces with Legendrian boundary

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.

math.DG↗
arXiv2019

Uniqueness of stable capillary hypersurfaces in a ball

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

math.DG↗
arXiv2018

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

math.DG↗
arXiv2015

Chern's magic form and the Gauss-Bonnet-Chern mass

In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implies a simple proof of the equivalence between the ADM mass and the intrinsically defined mass via the Ricci tensor, which was reconsidered by Miao-Tam \cite{MT} and Herzlich \cite{H} very recently.

math.DG↗
arXiv2013

A penrose inequality for graphs over Kottler space

In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.

math.DG↗
arXiv2013

The GBC mass for asymptotically hyperbolic manifolds

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\H^n$. In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $\H^n$ for any horospherical convex hypersurface $Σ$. As an application, we obtain an optimal Penrose type inequality for the new mass defined in the first part for asymptotically hyperbolic graphs with a horizon type boundary $Σ$, provided that a dominant energy condition $\tilde L_k\ge0$ holds. Both inequalities are optimal.

math.DG↗
arXiv2013

A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.

math.DG↗
arXiv2013

A positive mass theorem in the Einstein-Gauss-Bonnet theory

As an interesting application of the Einstein-Gauss-Bonnet theory and our work on the Gauss-Bonnet-Chern mass (Ge, Wang, Wu), we obtain a positive mass theorem for asymptotically flat graphs in $\R^{n+1}$ under a condition that $R+αL_2$ is non-negative, where $R$ is the scalar curvature, $α\in\R$ a constant and $L_2$ the second Gauss-Bonnet curvature. A Penrose type inequality is also obtained in the case $α>0$.

math.DG↗
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space $\H^n$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{equation*} \begin{array}{rcl} \ds \int_Σ\s_{2k}\ge \ds\vs C_{n-1}^{2k}ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 1k + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} \end{equation*} provided that $Σ$ is a horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $σ_{j}=\s_{j}(κ)$ is the $j$-th mean curvature and $κ=(κ_1,κ_2,\cdots, κ_{n-1})$ is the set of the principal curvatures of $Σ$. Also, an optimal inequality for quermassintegrals in $\H^n$ is as following: $$ W_{2k+1}(Ω)\geq\frac {ω_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(Ω)}{ω_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}}, $$ provided that $Ω\subset\H^n$ is a domain with $Σ=\partialΩ$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $W_r(Ω)$ is quermassintegrals in integral geometry.

math.DG↗
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_Σ\s_4 d μ\ge \ds\vs C_{n-1}^4ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 12 + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array}\] provided that $Σ$ is a horospherical convex hypersurface. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$.

math.DG↗
arXiv2013

The Sasaki-Ricci flow on Sasakian 3-spheres

We show that on a Sasakian 3-sphere the Sasaki-Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki- Ricci soliton. We also show the existence and uniqueness of gradient Sasaki-Ricci soliton on each Sasakian 3-sphere.

math.DG↗